If the coop is short then headroom is an issue to be concerned about. I don't have much of an idea of what your coop looks like (pictures help...hint...hint
) but you may be able to raise the roof a bit. You could incorporate some added ventilation if you need to. By removing the roof (this is where having used screws can be nice) you could scab on some upright 2x4's to the corner posts/poles thus extending the roofline. Set the roof back on top and secure. The gap between the old wall and the roof could be filled with wood or hardware cloth or both. By raising the roof you could put the flooring on top of the floor joists and have a deep layer of litter without losing head space for the chickens. I would make it a little taller than planned and with extra ventilation at the top for good ventilation that will be over the chickens' heads thus keeping them out of a draft during the winter. Remember, they need approximately 1 square foot per chicken of ventilation.
By the way, if you didn't build with screws then you would need to address the roof removal differently. But, that can be done, too. I won't go into all that unless you decide to raise the roof and need some ideas on it.
I think you're talking about the BlackJack roofing tar that some people coat the flooring with. They speak well of it...be sure to get it down into the cracks and crevices good. It would be good if you could paint the plywood edges good before installing it into the coop.
If you decide to try a single trapdoor on the bottom position the hinges so that the door swings downward parallel with the floor joists. This will help all the spaces between the floor joistsl to empty out rather than having one space that doesn't have much room between the bottom of the joists and the opened trap door, thus making it harder for the litter to fall out of it. Does that make sense?
How tall is the inside of your coop? What's the floor dimensions?
Btw, shredded newspaper should work fine ,if used sparingly. Once wet it tends to mat down pretty good. As long as it's mixed well with wood chips, dried leaves, dried garden waste, etc., and not over-used then it should be ok. There have been recommendations in the past not to use color pages but they should be safe to use...back in the day of lead typeset and toxic inks it was indeed a problem. I believe the industry has cleaned up this bit of pollution courtesy of the EPA implementing guidelines that prohibit heavy metals. Plus, using the shredded paper sparingly there really isn't a reason to risk colored paper if you don't want to.
Best wishes,
Ed