Restoring Files from Time Machine to new Mac

Hello, thanks for any help you can provide. It has been a bad week for my Macs. My early 2011 MacBook Pro is dead in the water (3 occasional beeps after start up attempt), it may or may not boot then crashes (thinking it is a hardware issue), and my iMac will not boot, possible hard drive failure (no apple in boot, white screen only then shuts off about 30 seconds into the boot), it crashed while I was using it. HD was replaced 2 years ago.

So...….I bought a new MacBook Pro, and assuming I can't get the PB to work in Target Disk Mode as my Thurnderbolt 2 monitor port is dead and I don't know if the Firewire 800 is working, I do have a fairly recent Time Machine Backup on an External HD. I do not know if I ever backed up the OS, but I think I did back up my applications.

What I am wondering, if I can restore files from the Time Machine and I do a restore all files and OS from time machine, will it try to over write the new OS? I want my keychains but I don't think there is anything else from the old OS I really need. most of what is in the old PB is backed up also on DropBox and iCloud, and both my macs are linked to those accounts, so those files aren't in bad shape. It is the personal movies, pictures and that sort that I am looking for that wouldn't be in either of those places.

Thank you