The beauty of pen-and-paper RPGs is that they aren't scripted the way computer RPGs are. The GM plays the opponents, so start thinking like someone who is defending his house.
Can a burglar break into your house, take a tv, and leave? Sure. Can he do it every day for a week? The homeowner is going to take actions to stop him. Maybe by setting up an alarm system, or hiring a security firm to post guards.
Nothing stops your player from going back to town after every encounter, but the bad guys will take advantage of his absence to bolster their defenses.
As for sleeping eight hours to regain HP, a character cannot get that benefit more than once a day. So resting five days and doing nothing else gets 10 HP healed, but the world moves on. Maybe another group of adventurers takes care of the monsters while the group was resting. Maybe the monsters decided there were easier pickings elsewhere, and moved on, taking their treasure with them.
As for the situation with the skeletons, I would rule that the skeletons are going to follow him, attacking all during his descent.
Enemies aren't static, like in computer games. That's a feature, not a bug. Make sure your player(s) know this, because it can be a rude awakening if you're used to computer games.
First off, welcome!
1. This particular quest has mats that you can sleep on. Every time he survived an encounter with a monster, he went back to these mats. Slept 8 hours and regained 2 HP, then slept 8 hours again to regain 2 HP again and repeated until his HP was full.
This seems like it could be possible, as the monsters in this area have all been defeated.
But it also seems like cheating.
Can someone clarify how this should work?
Even if the adventure is silent on this point, as a GM I would allow this once or twice before having one or more monsters respawn or perhaps I would start having a random chance of this happening starting the first night (say a 1 in 10 chance, i.e. a 1 on a d10) and then increase it by one every night. SOMETHING would happen; be it another adventurer, a cave-in, strange-noises, or whatever. In the end though, there really isn't anything wrong if the area is cleared and you allowed the PCs to heal up to full, it's all up to you as a GM or whatever all the player's agree to in a shared game.
2. The other thing he wanted to do between monsters is go back to town each time to spend gold.
He defeated all the monsters at the beginning of a dungeon, so he wanted to walk back out again and back to town.
Is that possible in the middle of a quest?
Absolutely. However, depending on how long an adventure leaves a quest/dungeon changes might (and probably should occur).
3. How many weapons can a character hold?
Can they hold a dagger in one hand, a sword in the other and have a crossbow on their back? Or a second sword on their back?
I think it comes down to weight and storage, but I’m not sure how all that works just yet.
Check out the section on ONE or TWO-HANDED weapons. That should help.
The Beginner Box really doesn't burden (pardon the pun) players with information about weight and storage, but to answer your question:
A PC could carry a one-handed sword in one hand, a dagger in the other and have a crossbow (and a quiver of bolts) on the back.
Unless a class feature or FEAT allows otherwise though, in the Beginner Box, even with one weapon in each hand a PC still only gets one attack each round.
4. He also had a potion of levitation and one for invisibility.
After going up a 20ft wall, he discovered 3 skeletons.
Using a rope he found, he wanted to go back down the wall, turn invisible, climb back up the wall and levitate over their heads.
What would an experienced GM say to that??
The short answer is "yes". You could certainly allow the PC to automatically spot the skeletons just as he gets up to the ledge and then allow him to quickly come back down. Invisibility would certainly help and should let the PC pass undetected and the levitation would allow the PC to move up to the ceiling and then pull himself along past the skeletons.
My longer response is:
Being the intro adventure that it is, the encounter in question simply says "the skelton's attack on sight", so unless the player said he was being particularly quiet/stealth, the skeletons probably should have attacked right off.
However, if you read the rules (and the more advanced rules later on), you will come across this situation/question many times: do one set of creatures detect the approach/movement/actions of another set?
This is handled by the "sneaking" character(s) making a Stealth check--probably with a -2 penalty in this particular case because of the climbing--and the creatures that might detect something making a Perception check.
So, step one, I would roll a Stealth check for the PC and a Perception check for the skeletons.
Step 2, when the PC reached the top, I would have the PC roll a Stealth check and then have the skeleton's roll a Perception check, but again if the PC were being careless and making a lot of noise or talking, then the skeletons would probably just notice the PC right away.