As you certainly know, I consider the three non-high PIE3 vowels to have emerged from a single PIE1 vowel, *a (which is of course unrounded), and it may well be that the antecedent of PIE3 *o was not yet rounded in PIE2, though I expect rounding to have happened already by that time. IMHO, in PIE2 *o probably was something like [ɒ] (and *e was [æ]). If PIE2 *o was unrounded, then the difference between *h2 and *h3 at that time would have been something else than labialization, but that begs the question why *h3 labiovelarizes an adjacent [ə] to /o/ in Proto-Greek if it doesn't have the feature [+labial]. One would have to suppose parallel rounding changes in both *o and *h3, which is not very parsimonious. The most parsimonious solution would IMHO be that *o was already rounded (and be it as [ɒ]) in PIE2.
But alas, we are probably not knowledgeable to find out when PIE *o acquired its rounding. Each of us has his/her own opinion about the time of *o-rounding, and we can leave it at that.
The Lycian fortition of laryngeals, I agree with you, is a development internal to Lycian. As I said earlier, there is no evidence for hardened laryngeals in Luwian, which was to be expected if the Lycian stop reflexes of the laryngeals were old.
Also, regarding the frequencies of the laryngeals in PIE, it may be that *h1 is heavily under-reconstructed (and the other two perhaps over-reconstructed). There are AFAIK quite a few words where a *h2 or a *h3 is reconstructed based on vowel colours alone without Hittite cognates actually showing the laryngeal in question, and in such situations, *h1, which doesn't colour vowels, would be irreconstructible. Also, there are the many laryngeals of unknown quality. How many of them were which one?
I do not seriously expect to manage to crack this riddle!
...brought to you by the Weeping Elf
Tha cvastam émi cvastam santham amal phelsa. -- Friedrich Schiller
ESTAR-3SG:P human-OBJ only human-OBJ true-OBJ REL-LOC play-3SG:A