Will I be awarded Ineligible quest score in borderlands?

So I started borderlands later than my friends and I am behind them almost 10 levels. They insist I play with them and finish the game early so that we can play borderlands 2 together.

I want to reach at the same point in story-line as them and complete the game together but I have a concern. Since I am playing Co-op session with my friends and complete quests that I am ineligible to do; When I play solo will these quests be completed for me or do I have to complete them again?

No they will not IIRC. You shouldn't be receiving rewards from them either.

I'd personally recommend using a save edit program (like WillowTree#) to get you up to par on quests/levels with your friends for now.