Riding outside of Lithuanian borders, you'd need to have some form of proof that you're allowed to use the bike. Depends on how Lithuania does it - could just be a hand-written note, ideally a notarized authorization, or your friend could add you to the registration as an authorized user (like they would do for a spouse). Chances are that nobody is ever going to check this, but it's worth doing the minimum for peace of mind.
For insurance, teeeeechnically you don't need the printout within EU/EEA borders - you just need to be able to prove that you have insurance. E.g. a PDF of the policy on your phone should suffice. But again, for peace of mind, best to go to the insurance agency's office and ask them to print out the actual Green Card on green paper. They are required to do this for free on request.
Oh, and you obviously need to have all the paperwork you possibly can to cross into Ukraine.
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