Which STATE!!??

1。从婚礼到现在已经6个月多一点了，还在有效期限里吗？

http://www.nolo.com/legal-encyclopedia/statute-of-limitations-state-laws-chart-29941.html
See limitation for written contract in your state.

2. contract是我和我老公两人的名字，当时付钱的是我老公，用的他支票簿。我可以以我的名义起诉吗？或者我和老公一起起诉，到时我自己出庭？因为老公工作比较忙，可能没办法为出庭专门请假。
Yes, you can sue, appear in court. Your husband doesn't have to go.

3. venue是owner私人的。我应该告owner还是venue？
Sue whoever signed the contract.  It could be a person or a company.  Upload the signature page of your contract and I can tell you. 

4. 我没有直接的证据证明我当时打扫了，venue也没损坏。但是当时有一个雇员确实是看着我们清理并且确认了的。现在联系不上她。这个要如何准备证据？应该我举证打扫了，还是他举证没打扫？
You are suing, so you have the burden of proof. If you don't have any evidence, you won't win.

5. 为了防止法院判我赢了他也不给钱的情况出现，我应该在法庭上做什么要求？
One step at a time, win a judgment first, then worry about executing the judgment. You can't ask the court to do ANYTHING before the court award you a win.

After your victory, you become their creditor (they owe you debt), and you can collect debts based on your state law.