Fekszik is an -ik verb, so its first person singular present tense indicative mood indefinite conjugation is "fekszem" (same as the definite one.)
No, they don't.
There's törik (to break (intransitive)), szökik (to flee), szokik (to get used to.) Most verbs conjugated from nouns with -ez/oz/öz, eg. borozik (to drink wine), babázik (to play with dolls or an actual baby), mozizik (to go to the cinema.) There's also -od/ed/öd, that usually gives an -ik too, such as nagyobbodik (to get bigger/larger), távolodik (to get farther away) &c.