Hey, 1: Yes you stay level 1
2:You grow stronger through Equipment and special"events". There are many options to get stronger Equipment in the City also there is stuff like training on the dummy in the hut of the hermit a few times to get 5 Attack.
3: In your case there are still 3 fights in the dungeon and you can either load a file of before you got inside and get better equipment or try fighting the same battle like 20 times until you get lucky enough to win.