(single) Patient Chart / Patient's chart
The first is a compound noun derived from the nouns patient and chart. As such, primary stress is on the first part of the compound: "patient chart". The second is a NP (noun phrase) derived from the determiner patient's and the noun chart. As such, primary stress is on the last part of the NP: "patient's chart". The possessive determiner patient's is derived from the NP patient and the ending 's.
It would also be possible to have a compound noun derived from plural noun patients and noun chart: "patients chart".
(plural) Program Characteristics / Program's characteristics
The first is the plural of the compound noun program characteristic, with primary stress on program. The second is a plural NP with plurality marked by making the head noun characteristic plural by adding the ending s, program's is a determiner, and primary stress is on the last element of the NP: "program's characteristics".
There are no adjectives in your examples. Some people sometimes call the first noun of a compound noun "adjectival", but it isn't really an adjective. Adjectives are part of NPs, not compound nouns.