Yes it does get better - honest.
We've been through similar awkward waking times with Pip. In the end we just decided to keep a spare duvet downstairs and dh dozes on there until he has to get up at about the same time you do.
What time is he going down to sleep? Any chance you could keep him up an hour later? Also, are you lifting his water bowl once it starts getting late, it's not something I normally recommend, but as long as he drinks plenty in the day it may help, as he might not be as desperate for a wee.
Re training. Some days are just like that. They are just babies after all. If he's not into a training session, just ditch it and try again another time.
If he's bored of his toys, have you by any chance a cardboard box you could give him. Either put his toys in there and give it a rattle or drop some treats in and let him at it. Pip has had two cardboard boxes for a week (well actually they are just pieces of cardboard now) and they've kept him occupied for ages chomping on the card, ragging the pieces and sliding across the carpet on them. 
If he's really bitey, take a short walk up the road for socialisation, it soon brings them down a peg or two when they are reminded how big the world really is and they tend to sleep for a while afterwards while they process the enormity of it all.
Also, try not to be so hard on yourself. You don't have to be totally focussed on him 24/7, he needs time to chase the odd leaf and entertain himself as well. It's actually just as important a learning process for him as the formal training.
Can't help with the cat thing though - we don't have any, because dh is allergic to them. Sorry he/she drew blood though, I can imagine that was the last straw.
Hope you get chance for some R&R tonight and he sleeps a bit later for you in the morning.
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