Does Jesus pray for everybody?

Not according to Jesus He doesn't.  I don't think He can make it any plainer than that.

John 17:9(NKJV)

“I pray for them. I do not pray for the world but for those whom You have given Me, for they are Yours. 

Or to put it another way, Jesus only prays for those who have accepted Him. 

Now don't confuse what Jesus does in this situation with what He instructed us to do.
 

Matthew 5:44(NKJV)

But I say to you, love your enemies, bless those who curse you, do good to those who hate you, and pray for those who spitefully use you and persecute you, 

Does that seem contradictory?  I don't think it is and I'll tell you why.

As far as who Jesus prays for I think that makes total sense from the standpoint that we are described as heirs with Jesus when we accept Him.  I see being prayed for by Jesus as one of the things that we receive because we are heirs that non-heirs are not entitled to.  Makes perfect sense to me when I look at it this way.

As far as our enemies, I think the point here is that we should continue to pray that they come to accept Jesus.  That is how we can demonstrate our greatest love.  Does that make sense?