Lawsuit Against a City

Depends on how it is addressed in the city charter and what powers it gives to whom and for what purpose.

As long as its done openly and publicly with signed receipts i should think it'll be alright

Most municipalities have legal counsel who would advise the city council.

I doubt if a mayor, by himself, can settle anything without bringing it to the council.

But I suppose it would depend on the amount and terms of the settlement.

Well, i don't think that mayor, by himself, can settle anything. For doing this he should bring it to the council.