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In the sentence –

Auditors recommend an increase of the allowance for bad debts by ten %.

– should the percent sign be there or should the word percent be spelled out. Also should the number be written in a numeral format?

In general, it is good practice that the symbol that a number is associated with agrees with the way the number is written (in numeric or text form). For example, $3 instead of 3 dollars.

Note that this doesn't apply when the numbers are large, so it is perfectly fine to write 89.5 percent, as eighty-nine-and-a-half percent is very clunky.

This source puts it simply:

When writing percentages and money references, use the numeral with the percent or dollar sign.

Also, this answer on Academia.SE says "APA version 6 style manual has an entire section starting from 4.31 on how to present numbers in text" and provides a summary.

Therefore, you should write either 10% or ten percent - note that it is a common mistake for some to write per and cent separately. For some classical humour, check out this xkcd post.

What is incorrect is for the number to be written in text form but not for the symbol. That is, ten % or £ five is not good phrasing.

If you're writing out the number then write out 'percent', e.g. ten percent, otherwise use the sign, e.g. 10%.

Style guides differ considerably on the best way to express a percentage such as ten percent. To make matters even fuzzier, some of them have different advice depending on whether the term appears in running text or in a special element (such as a table) and depending on whether the subject of the text is general or heavily scientific or statistical. Here is a quick survey of different style guide advice on how to render percentages.

From The Chicago Manual of Style, sixteenth edition (2010):

9.18 Percentages. Except at the beginning of a sentence, percentages are usually expressed in numerals. In nontechnical contexts, the word percent is usually used; in scientific and statistical copy, the symbol % is more common.

Fewer than 3 percent of the employees used public transportation.

With 90–95 percent of the wok complete, we can relax.

A 75 percent likelihood of winning is worth the effort.

Her five-year certificate of deposit carries an interest rare of 5.9 percent.

Only 20% of the ants were observed to react to the stimulus.

The treatment resulted in a 20%–25% increase in reports of night blindness.

Note that percent, an adverb, is not interchangeable with the noun percentage (1 percent is a very small percentage). Note also that no space appears between the numeral and the symbol %.

As its examples show, Chicago endorses "10 percent" in most general-text situations, "Ten percent" at the beginning of a general-text sentence, and "10%" in scientific and technical text. Chicago doesn't address how to handle "10%" if that term were to appear at the beginning of a sentence in a scientific or statistical text, but I imagine that it would advise you to use "Ten percent," with the further proviso that you should recast the sentence to avoid putting the percentage term at the beginning if the spelled-out form seemed excessively awkward (that is, it would recommend changing "Ninety-seven point three percent of survey respondents said..." to something like "In the survey, 97.3% of respondents said...").

From Margery Fee & Janice McAlpine, [Oxford] Guide to Canadian English Usage, second edition (2007):

percentage, per cent, percent Percentage is always one word. Canadians prefer the two-word spelling of per cent, although the single-word spelling is also common. British dictionaries list per cent first, American dictionaries percent.

Per cent is followed by either a single or a plural verb form, depending on the related noun: 'Fifteen per cent of the total is added to the bill for service' or 'Only twenty per cent of the students are going to pass the test'.

The per cent symbol (%) is used only with figures: '8%'.

The examples suggest that Fee & McAlpine would support one or the other of the options "ten per cent" or "10%," depending on context; and they explicitly reject use of the mixed form "ten %." However, the mixed form "10 per cent" goes unmentioned.

From The Associated Press Stylebook (2007):

percent One word. It take a singular verb when standing alone or when a singular word follows an of construction: The teacher said 60 percent as a failing grade. He said 50 percent of the membership was there.

It takes a plural verb when a plural word follows an of construction: He said 50 percent of the members were there.

Use figures: 1 percent, 2.5 percent (use decimals, not fractions), 10 percent.

AP evidently opposes use of the percentage symbol under any circumstances and allows only the form "10 percent."

From Bryan Garner, Garner's Modern American Usage, second edition (2003):

percent; per-cent; per cent; percent.; per centum. This sequence illustrates in reverse the evolution of this word, originally a phrase. Today it is best spelled as a single word. The plural of percent is percent; adding an -s, though not uncommon, is substandard.

In most writing, 75% is easier to read than 75 percent or (worse yet) seventy-five percent. Prefer the percentage sign when you can. Many styles, however, insist on spelling out percent.

...

...Writers must be careful with percentages and percentage points. For example, if the unemployment rate rises from 4% to 6%, both of these statements are true: Unemployment is up two percentage points, and Unemployment is up 50%.

Garner's is the only reference work I consulted that strongly recommends expressing percentages in the form "10%," regardless of the type of text one is dealing with.

The Oxford Guide to Style (2002) takes three swings at the subject:

3.2 Usage

Abbreviations in running text

Prefer nineteenth century to 19th cent., and 25 per cent (two words, no point) to 25% in text; rules for notes and tabular or parenthetical matter differ.

...

7.8 Statistical texts

...

The symbol %, rather than the spelt-out per cent, is permissible in text as well as in peripheral matter such as tables, notes, parenthetical material, and captions.

...

13.9.1 Use of scientific style [in the social sciences]

Since social science as a discipline resides somewhere between the arts and the 'hard' sciences, a decision must be made for each typescript whether a scientific style should be followed in presenting the text and references. The higher the level of statistical material—as evidenced in economics, econometrics, demography, geography, and the like—the more appropriate scientific style becomes. In text this usually manifests itself in such matters as spelling out numbers up to ten only (rather than up to on hundred), and using % rather than per cent.

Oxford's scattershot coverage—and lack of relevant examples—might lead one to suppose that it countenances "ten %" in scientific texts; but I suspect that the juxtaposition of "spelling out numbers up to ten only" and "using % rather than 'per cent'" is an unfortunate accident, not a recommendation to use "ten %" in social science texts. The only example relevant I could find in this style guide involves parenthetical use of "74%" and "26%" in statistical texts—which doesn't address the specific case of "ten per cent" (Oxford does say that "Figures in scientific and technical work are usually spelt out up to nine," which would be a vote for "10%," but that doesn't help its users figure out how to handle, for example, "nine per cent").

From Allan Siegal & William Connolly, The New York Times Manual of Style and Usage, revised edition (1999):

percent, percentage. Percent is one word. The preceding number is always expressed in figures (except where it begins a sentence): 80 percent; 8 percent; one-half of 1 percent; four-fifths of 1 percent; 0.5 percent. But: five percentage points; 12 percentage points. The symbol % may be used with a figure in headlines, tables and charts: 5% Raise; 93%. Do not use the abbreviation pct.

As this guideline indicates, the New York Times follows the usual U.S. style guide preference for "10 percent" in running text and "10%" in special matter (such as tables and headlines).

From Joseph Gibaldi, MLA Style Manual, second edition (1998):

3.10.4 Percentages and Amounts of Money

Treat percentages and amounts of money like other numbers: use numerals with the appropriate symbols.

[Relevant examples;] 1%[;] 45%[;] 100%

In discussions involving infrequent use of numbers you may spell out a percentage or an amount of money if you can do so in three words or fewer (five dollars, forty-five percent, two thousand dollars, sixty-eight cents). Do no combine spelled forms of numbers with symbols.

MLA thus explicitly rejects spelled-out numbers and the percentage symbol (such as "ten %").

From Words into Type, third edition (1974):

Percentage. In literary works percentage numbers are spelled out: "fifty percent." In technical and scientific writing, numerals always precede the word percent, with the single exception that isolated references one percent may be spelled out. In statistical material and where other numerals appear frequently with abbreviations in scientific copy, the percent sign (%) may be used in text matter. Always use the sign (%) in tables, and never use a fraction before the word percent or the percent sign.

10 percent[;] 2.25 percent, not 2¼ percent[;] 12.3 percent[;] 0.5 percent, not ½ of 1 percent

Words into Type is considerably older than the other style guides I've cited here. It endorses the spelled-out form "ten percent" in what it calls "literary works," but insists on "10 percent" in technical and scientific copy and on "10%" in tables.

Conclusion

As the preceding excerpts indicate, style guides are all over the map in their preferred formatting for percentages. Depending on where you live, what your subject matter is, and which style guide you follow, you may find that the recommended form is "ten per cent," "ten percent," "10 percent," or "10%." I could not, however, find any style guide that explicitly endorsed the form "ten %."