Okay, so. You want to create a standard invoice that your client ("Rechnung+ Client") will send to her or his client ("Client X"). Is that right?

In answer to your first question: yes, remove the bank account information and you can use it as-is. You might want to change the VAT ID line to "EIN" which stands for "Employer Identification Number" -- essentially this is the number a client would use to report to the federal income tax authority the money they paid you.

And, in answer to your second question, you don't include the check when you send an invoice to your client. You simply tell them "this is what you owe" and, if relevant "these are my terms"*. It is the client's responsibility to cut the check and send it to you or to arrange to transfer the money, and it comes from their bank, not yours.

-Sarah L.

*I would consider leaving a place for "Terms" on the invoice. I've been in a discussion on another site about handling clients who are reluctant to pay you--if you don't state the conditions of the sale, you can't take advantage of them. So, whether or not you include such things in the contract, if any of the statements below are relevant they should appear on the invoice:
1. Payment is due on presentation of invoice.
2. You may take a __% discount ($XX) if paid in Y business days.
3. We guarantee the quality of our work. Please contact us first if you have questions or problems.
4. All payments subject to a ___% service charge after 30 days.