There's no limit on the number of times you can explore as such...
If you win/lose an encounter it doesn't matter. So long as you have a card/power (or location effect like General Store) which lets you explore again then you can go again.
Note that if you beat a Villain then your location closes and (usually) all the cards are banished so you cannot explore any more in that circumstance but that's only because you cannot move and have an empty location deck where you're standing.
You can discard to explore again multiple times. Similarly, the General Store lets you explore multiple times (e.g. if you find a monster -> explore again and find a monster -> explore again and find a monster -> explore again and find a monster -> etc..) If the whole General Store location deck is just one Monster you can encounter it an infinite number of times (happened to us last night)!
So, in summary - YES to all of these :)
Your sense is most common.