[RPG] How do hit points work

The magic item you're thinking of is probably the Potion of Regeneration. The level 9 version costs 160 GP a pop and gives you regeneration 5 while you're bloodied. It also costs a healing surge, which might be a problem for your wizard if you do a lot of encounters in a typical day.

Trollhide Bracers have a daily that also gives you regeneration 5, not just while you're bloodied, but those are a level 19 item so it'll be a while before you might have those.

Armor of Sudden Recovery is great for regeneration. You can, as a daily, convert any ongoing damage you're taking to regeneration -- so if you're taking ongoing 10, you can turn it into regeneration 10. I would strongly recommend a set. A slightly cheesy but amusing option: combine that with Dragonfire Tar, an alchemical consumable which does ongoing damage on the target. Use that on yourself. Check with your GM before using this one, though.

Back to non-cheese. For your neck slot, an Amulet of False Life has a daily that gives you temp hit points equal to your healing surge value. That's effectively an extra second wind once per day. I think that's better than the Amulet of Life, which allows you to spend an extra surge once per day when you spend a surge, since you have to wait for someone to heal you or use second wind with the Amulet of Life.

Diamond Cincture is another effective second wind, once per day. So that's pretty good. It fills your waist slot.

OK, let's see. Strategies -- don't forget about Shield. That's a very good defensive utility. With any luck it'll prevent one hit per encounter. For your level 10 utilities, you have a few interesting choices. Mass Resistance is a life saver if you're fighting anything at all elemental. It's party-oriented, which is nice. Blur is a great power to protect you -- you can't be seen by enemies more than five squares away, plus you get some defensive bonuses. If you have some way to fly, which you probably will eventually, this is a great way to stay safe in the air.

You were correct -- by the book, you can absolutely use encounter powers outside an encounter. Healing powers are the most common example of this, but there are others -- for example, the bardic power Words of Friendship, which gives you a boost to Diplomacy, is almost always going to be used in non-encounter situations. As @veritascitor points out, an encounter power is just a power that can be used once every five minutes in non-stressful situations.

Bear in mind that if you use Healing Word after a short rest, which is a must for Restful Healing, you don't immediately recover it. You'd need to take another short rest to get it back. Sometimes you'll have another five minutes to spare; sometimes you won't.

Something else that might come up in this regard is the duration of powers that last until the end of the encounter. Page 278 of the Player's Handbook covers this: they last "when you take a rest (short or extended) or after 5 minutes."

Finally, if your group has a house rule that says you can't use powers outside combat, well... then they have that house rule. I'd point out the example you noted, and cite Words of Friendship, but in the end every group does have the right to come up with their own variations on the rules.