Your chrome will at minimum tarnish closer to the engine. But that's probably the least of your problems. Sink traps and chromed drain pipe are all made of a fairly light brass alloy, and they don't withstand a great deal of abuse. They're fairly easy to bend, they crack fairly easy, they crush fairly easy, the chrome peels off of them without too much trouble.
As far as soldering them, there is a torchless solder out there. I haven't used it, but I have seen it, and everything about it suggests it would work. Thus if you wanted to try it, you could stick it all together without tarnishing your chrome until you started it!
Buy a stick or 2 of each and find yourself a 1 1/4" and 1 1/2" sweding tool so you stretch that pipe up big enough to size it up and solder it.