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I am very confused of the grammars behind the phrases or words like problem finding, problem solving and pain killing.
I feel that rather than present participle, past participles should be used because the proposed nouns, like problem and pain, are the objectives of the verbs, like find and kill.
Your examples show two very different constructions, so the answers are different in the two cases.
"Pain killing pills" (with the participle, or verbal adjective, modifying the head noun "pills") can be paraphrased as "pills which kill pain". The participle needs to be active, which in English gives no choice but the present participle (the past participle is passive, or middle in sense). "Pain-killed pills" is grammatical though unlikely, and would mean "pills whose pain had been killed"
"Time consuming work" is the same, and might be paraphrased as "work which consumes time".
But "skills in problem solving" is quite different, because "problem solving" is a noun phrase, and "solving" is not a participle but a gerund, or verbal noun.
The “verb part” does not have a tense or aspect at all. It’s the gerund form, where the verb is used much like a noun; it just happens to have the same -ing ending as the present participle. The structure is essentially the same as the appositive—e.g., a tennis player is a player of tennis. With that interpretation, the meanings become quite clear.
These are pain-killing pills.
These pills are for the killing of pain.
This work is time consuming.
This work is characterised by much consumption of time.
He has good skills in problem finding and problem solving.
He has good skills in the finding and solving of problems.
It's to do with the "completeness" of the actions in question. Almost like an imperfect action, it is or was an on-going process. As far as I'm concerned, at least.