No, the owner still decides how to spend the experience on this player.

A player you loaned to another team will still show up in your training complex.

I've added this bit of information to the help.

What is the daily experience the loaned player receives? Is it from the training center of the owner or is it the from the training center of the club he is loaned to?

Most realistic would be the xp of the training center of the club he is loaned to, as that is the place where he will train.

I have put two players for loan. Can you please tell me if they are visible and available for bid?

Is there any limit of transfers for each "transfer window"?

No, there is no limit on the number of transfers you can do. Though I suppose there is a practical limit of what is possible in this time.