The most idiomatic usage of 'free of charge' is at the end of a sentence. "I'll allow you to stay free of charge." "The cigars are free of charge." The phrase is rather clumsy outside of advertisements.
In place of 'free of charge,' there are single word alternatives with equivalent meanings. Complimentary and gratis work well. The word 'free' is also generally an acceptable alternative, although somewhat ambiguous. 'No-cost' also works well (suggested by Jim Reynolds in a comment to this answer).
As for your questions:
Do you write or say other adjectives before or after the adjective free of charge used together with a noun? If forced to use free of charge in combination with other adjectives, I would place free of charge last. Severely cumbersome adjectives such as this should be handled as such (or, more preferably, alone).
Is it better to put other adjectives before or after free of charge? Same as above.
Do you put an indefinite article before free of charge? Yes, see below.
How often is free of charge used with an indefinite article? Surprisingly (to me, at least), "a free of charge" is a more common phrase than "the free of charge." Thus, the indefinite article is more common than the definite article before 'free of charge.'