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1。如果继续驱赶， 是否有法律依据？对方支票后来寄出了。

You will stop eviction once the check is cleared.  And they still might owe you late fees.  

2。如果让他们住， 要不要签什么条款，以防后患 （狗死的问题）。

3。 要不要退宠物押金？

These two issues can be combined together. 

Pay attention to the rule: https://smallbusiness.chron.com/contract-modification-procedures-44003.html "Under common law, a party to a contract needs new consideration -- something of legal value -- to modify a contract. The party obtains new consideration by negotiating with the other party to change the consideration on both sides of the contract. "

New consideration means: a contract can be modified only when both sides entered into a new bargain that benefits both parties.  A modified contract is invalid if one party is carrying out preexisting duty under the old contract and the other party is getting something new in the modified contract. 

So, two of you had a lease.  If you simply had them sign a new lease with less deposit and less rent, the result is landlord is performing "preexisting duty" in old lease while tenants get something for free (lower deposit and rent).  This modified contract is not valid. 

Both of you have to get something out of a new contract. 

So it is reasonable to ask for a new lease, that they will have lower deposit and lower rent, while you get a new clause in the lease that no pet is allowed.  This is just a example of new bargain, what actual clause you might get is up to your negotiation and their desires.