I think your pup is still developing control of his bladder and this will sort itself out in time - keep doing what you are doing with the house training and it will come right in the end. The fact that he only does a small amount in his bed before moving to another area suggests that he can't yet anticipate when he needs to go until the last second. His behaviour then is normal - move away from the bed.
The other main possibility is that he is toileting normally away from his bed and the small amount of urine in his bed is done when you arrive home through excitement. If you think this is most likely try ignoring the puppy when you get in until he is calm instead of greeting him straight away.

I am concerned that you are using bleach - this can cause damage to your puppy. It can burn his skin, irritate his eyes and lungs and cause internal damage if he eats it when chewing up the newspaper. Paper on its own is enough.

I am not an advocate of crates and don't think that it would help your pup if his problem is one of the reasons above.
When you are at home all day try to lengthen the time between toileting gradually. Try an extra 15 minutes at a time, this will help the pup practice holding on until he is outside.