technically, the rules state that you can ride if you have a park hopper ticket. BUT, in all of my trips, I have never been asked for ID to ride. Some folks have stated that the resort monorail has asked for resort ID during peak times.... your mileage may vary....
We've been there a lot, see sig, and have never been asked to prove we are staying on property, even though we always do. May I point out that the people going to the Magic Kingdom, and parking their car in the lot, can use any of those modes of transportation to get to the MK. ( Well, maybe not the buses )
Yes Your admission tickets give you use of WDW transportation. Keep in mind tho there's no direct transportation by the WDW DTD busses to the parks. You'll need to take one to a resort and transfer there for a park. If you want to go direct use your Hotel's transportation.