From my experience in Germany as a part-time freelance software developer with a full-time job (also as software developer), there are a few things to observe:
First of all check if your employer agrees about you taking additional work on the side. They're liable if you work over 10 hours a day and that could happen with a side job they don't know about. Your employment contract is a good place to start.
Your income from freelance should be less than what you make from employment or you'll be in for additional health insurance costs. Ask your health insurance company and they may tell you a number. Freelancing should not represent your main source of income.
Accounting for freelancers (Freiberuflich) is not as tough as for companies (GmbH, AG), but still involves monthly work (Umsatzsteuervoranmeldung) and yearly declaration of profit (Einkommenssteuererklärung). The intervals may be extended once your first year is past (quarterly or yearly declaration).
About your specific questions:
Clients expect a bill to satisfy all legal requirements. The exact wording of a bill depend on your legal status (e.g. freelance differs from small company).
If you resign from your company, don't expect any unemployment benefits at all. I did that when going full-time freelance, wrote a business plan and was right out rejected at the unemployment center. You won't get any help from them unless you mean to start a company with employees or you lost employment without just cause.
I'd recommend to look for a seminar on starting a company (Existenzgründerseminar). You can find them in many places (like Volkshochschule) and it's a great place to start to understand how a company is run (at least in theory).
Generally, for any kind of payment you receive as freelancer, you have to give an invoice. And in order to be able to do that, you need some sort of legal form, either single-person company or a full company.
The client will always need an official document (invoice) to justify their payment (money going out of their company). If a client tells you he/she doesn't need an invoice, that most likely means they plan to pay you "under the hand"/off the books.