Learn Turkish: Kafamdaki sorunlar yavaş yavaş halloluyor. - The problems inside my head are slowly being resolved.

It is a compound verb made from the noun hal (solution or state) and the auxiliary verb olmak (to be or become). When combining these, the final consonant of the loanword hal doubles. Because this sound change (consonant gemination) occurs, Turkish spelling rules require the two words to be joined together as hallolmak.

Could I use hallediyor in this sentence instead?

No, because halletmek is active and means "to solve" or "to handle", which requires a person as the subject. Since the subject here is the problems themselves (sorunlar), you need the intransitive form hallolmak, which means "to be solved" or "to resolve themselves".

What role does the -ki suffix play in kafamdaki?

It turns the location kafamda (in my head) into an adjective that directly modifies the noun sorunlar. It allows you to specifically say "the problems that are in my head". Without -ki, the sentence would simply mean "In my head, problems are resolving".

Why use the repeated phrase yavaş yavaş instead of just the adverb yavaşça?