Looks like the training is going well.
Adorable pup.
Quick suggestions to help you out a little...
-using one word commands like down instead of lay down often are easier for the dog and the actual action from the dog then is also quicker
- if he doesn't do it on the first time you say the command put him on lead or coax him with food a hand signal or toy instead of repeating the command. Repeating can sometimes actually teach them to ignore you and do things when they feel like it and not when initially asked.
- use a different tone in your voice when doing a command then when giving verbal praise so there is a more definite difference
- when giving the praise immediately when the correct behavior is done mark it with whatever you are saying a little faster so he knows exactly what you are saying good boy about. (a clicker can help with this to make the split second it happens as well as just your verbal timing)
Keep up the good work