I would definitely be worried about bigger predators following the rats - that is exactly what happened with our mouse problem: I faced off with a venomous snake in a corner (not intentionally) last summer and removed a bunch of non-venomous snakes, then we had the signs of a weasel last week
Quick action goes a long way - I would remove food and deny access right away, as much as you can
I am still recovering from the enormous effort of last week - did everything as quickly as possible, too
I am no rat expert, but I know people pretty well - in my experience, we often overreact and make mistakes that can be fixed with some thought and swift action - I would still be hesitant to torture the wildlife for my mistakes, rather fix the mistakes and let nature take its course again - no animal is going to hang around if there is no food or shelter
maybe your neighbors really let things get out of hand in their shed - nice of you to offer to help - that may go a long way in solving the problem - if they have food in the shed and do not go there a lot, the animals probably love that and take advantage of the opportunity for them - God knows that's what humans do, given half a chance to use their environment to their benefit
definitely would be concerned about the rats moving into my house/garage/garbage/compost next and would try to think ahead and block the spread - we are after all supposed to be the smarter ones
"humankind - be both!" - bumper sticker quote, not sure about the author