IMO when you use something as an improvised weapon it is, ipso facto, not a weapon. Otherwise you’d apply your proficiency bonus, no? And if it’s not a weapon then it’s not a heavy weapon.
This is flatly not correct. You don't add the proficiency bonus because you aren't proficient, not because an improvised weapon is "not a weapon". You can gain proficiency by taking the Tavern Brawler, and then you do add your bonus, so your logic here is just totally off. Sorry.
Also, if ranged weapons lose their properties when used to make a melee weapon attack and melee weapon lose their properties when thrown, then not only can a halfling throw a great axe 20 feet without disadvantage, they can do so one-handed (dealing only 1d4 damage on a hit, of course.) YMMV on if that strains suspension of disbelief, but it certainly does for me.
Ehhhhh I don't think that's too bad, because they're clearly just lobbing it - it's no worse than a Halfling throwing any other bulky, 7lb+ object. If that strains your suspension of disbelief, presumably a Halfling throwing a chair, or a small cask of ale/whiskey/etc., or a bag of flour or whatever also does.
What that means though isn't that it's wrong for a Halfling to be able to lob a greataxe specifically, it means you are gonna need house rules re: small people lobbing heavy things.
Maybe just halve the range if you don't meet a certain STR/weight ratio?
Just as an aside, I don't think a longbow actually has the Heavy trait because it is physically heavy, but rather hard-to-draw. So even if as a DM, I ruled some weapons retained some traits, I wouldn't rule a longbow retained the heavy trait. A heavy crossbow maybe, because you could really clonk someone with one of those things (and it's hardly broken to allow GWM with a weapon that is likely non-proficient, and only does 1d4 damage, I mean, ooooh, I'm so scared of the Fighter with GWM getting a -1 to hit and doing 1d4+4+10 damage, instead of +2 to hit and doing 2d6+4+10 (for example).