I'm not so clear on the use of the dagesh as a point of definition -- I understand some fo the pronunciation issues but not all of them either so I would appreciate any help here.
I recall learning years ago that a dagesh in a final hei means two things -- 1) that the letter is pronounced in an aspirated way, and 2) that it refers to possessive.
I am wrestling with the second notion. There seem to be many different situations where the dagesh is found -- in some, the hei has no vowel but follows a letter with a kamatz (as in the second line of eishet chayil "batach baH lev ba'laH" and this seems to show the possessive in 2 ways:
baH = "in her" ba'laH = "her husband" (other words in the poem, like kapaH and neiraH seem to work like this as well).
However, in the rosh hashana davening, we say the word "vayigbaH" which has a hei following a patach and having a dagesh -- with no possessive indicated.
Even more confusing, there is also the case where the dagesh is in a hei which HAS a nikkud. The one that really jumps out at me is the word "elo-Ha" (or is it "elo-aH). In the rosh hashana davening, we say "v'ein elo-Ha mibal'adecha" which is not possessive, while on Yom Kippur, we say "v'al kulam, elo-Ha slichot" which means "the god OF" or "the god to whom belongs forgiveness."
So I don't see any consistency in the use of dagesh there in terms of possession and also meaning. This leads to having the word "elo-Ha" mean both god and god's. Is that accurate? does the dagesh have some other use or rule that I can use to understand why it would or would not appear?