(Native speaker here, only amateur linguist (-: ) Interesting question, linguistically. The short answer is no. In עוגה indeed the stress is on the last syllable, but in גובה="height" the stress is on the first.
But digging deeper, in quickly thinking I found very few words with a final ה that are stressed on the penultimate syllable. And in all those words, the ה is part of the radix. In the vast majority of words ending in ה it's not part of the radix but a suffix. It's possible that any such suffix moves the stress to the last syllable. But I'm not sure that's the best explanation.
To compound the situation, there are some words with final ה that "by the rules" are stressed on the last, but in spoken Hebrew are stressed on the penultimate: כמה, איפה, and many many given names: משה, חנה, שלמה. Might be that the penultimate stress is a left-over from Ashkenazy pronunciation.
More common not to pronounce the /h/ - given that in most contexts the "she" is not emphasized. Then again, if you "declare" it like the speaker here, you almost have to pronounce some consonant. Some speakers would tend to use the glottal stop א, but that would mark them as uneducated. So people would say /hee/. Sounds to me like in the recording here it went missing for some technical issue in the recording.