Yes on both.
Your designs save automatically to your account, so you can close the tab and pick up right where you left off.
Every room is included — your $49 fee covers your whole space, so you can twin and design as many rooms as your home has at no extra cost.
There are no per-room fees and no separate "project" purchases. Continued access to your saved spaces requires an active account in good standing — see "Do I need an active account to access my twin?"