Yes. There are always fees associated with refinancing.

Yes, any terms of a modification agreement are pretty much up to the lender. You do not have to accept what they offer. I do not know if they will consider a counteroffer that does not contain any additional contribution from you, but there is no law against asking.

They are basically allowing you to change the original contract, so they have the position of strength in this. However, they run the risk of losing your account to another lender (if you are a "good" account) or having the account go into default if you can't manage the modified terms and ending up foreclosing (which they don't want to do; they have no desire to own property if they can avoid it).