Hi, I have the quinny zapp, we got it when lo was 10 months old, he never had a problem with napping in it and he didnt lol forward at all, he is 22 months now so doesnt sleep in it any more.
I didnt have a bag for the underneath as they were expensive and tiny and I managed fine without.
I did have the travel bag as it came as part of the package I bought but tbh it was crap, the buggy only fitted in it if you took off the wheels and took the cozy toes off, I ended up braking the zip on my bag so in my opinion unless it comes free with the pushchair I wouldnt bother.
Not sure about a buggy board as I never needed to use one but I doubt it would fit as it doesnt have a bar at the back.
Just incase you are worried about the sleeping position here is a piccy of Isaac asleep in his.