Q&A, yay XD Mind you, I'm still an okay Heimer, there's a lot I still need to learn how to do with him.
What's his power fantasy; what's the thing you want to be able to do better than anyone else when you pick him?
When I pick Heimer, I pick him for a safe laning phase. There's comfort in knowing that I'm likely not going to die and can farm safely to the point where my farm is so good that I can transition easily into mid game without needing any early game kills to get there.
Though, if the jungler is squishy (mostly in my games with him I've been against Lee Sins, because Lee Sin is Lee Sin) and tries to gank me, they realize too late that the little lasers do a lot of damage and, in the case of Lee Sin who uses his gap closer to get into the laser ring o' death, finds himself dying fairly quickly as he tries to run out. Mid-late game, mix the lasers with 
and it's very effective.
However, if I'm in an unfavorable matchup that makes my laser turret things useless, then I'm screwed over. Of course, there's another poke-centered playstyle you can play Heimer with in that situation, but I'm not very good at it and that's not how I want to play him.
What's the most fun way to play him?
For me, it's the lane freezing farm fest in mid lane. Mostly because I've been through enough abuse mid lane and it feels really good to get back at the oppressive mid laners by denying them farm. And if I force them to roam, then they will lose their tower. Heimer definitely has an advantage over the melee mid laners, but he's not particularly bad in most matchups.
Then after the laning phase, Heimer can either stick with the team for pushing, splitpush by himself, or defend easily. His turrets also make securing dragon and baron very easy.
I've been experimenting getting 
and splitpushing two lanes for days, in which the portal goes, for example, bot lane, and then I go top while my team pushes mid. I dunno, maybe pushing and seizing towers is boring for most people, but that's what wins games XD
In what scenario do you think he is the least fun to play against?
If you're melee, the turrets will suck and you'll be farming under turret.
Likewise, if you don't know what a Heimer is or what a Heimer does, then it's easy for him to exploit you.
There's also an added bonus playing as Heimer as visually seeing the range of the turrets, so you can bait enemies to the edge of the range and they'll receive damage. Stunning enemies at the edge of the turret range is also effective since they take the turret damage along with the E damage and they can't do anything about it for a second, though the turret lasers are slow and can be dodged when the stun wears off.
Did anything feel unintuitive about him when you started playing him?
His W is the hardest skill for me to control. Regular W is an X-shaped skill where he fires a bunch of rockets at the cursor location. Or rather, they converge at the cursor location and then fire off in the opposite direction after the converge point. So, at max range all those rockets can hit one target for a snipe attack, and at min range it's more of a weaker AoE attack (still applies on-hit effects, though). Though it's still a skillshot and easy to miss/easy to dodge, and when you have quick casting turned on it can be confusing to figure out. Also it's easy to miss. Very easy to dodge. I miss this skill all the time. .-.
But then the Ultimate W doesn't do that converge thing (I've tried), it just goes straight in front of him in whatever direction you're pointing.
Then his E has a weird hitbox. Or maybe champions have a weird hitbox. Either way, his E is supposed to stun enemies if they're at the center of where it lands, but they're slowed if they're within the range of impact. In which I could've sworn the animation would show my bomb thing landing directly on the enemy, but they would be slowed instead of stunned. The Ultimate E just throws the bomb and it bounces three times in a row, with the same properties as the regular E.
Are there things he's good at that you'd be willing to give up if he was compensated in other, appropriate ways?
I'd happily sacrifice some of his turret-laser-thing stats for a passive that was actually meaningful. His current passive increases the HP Regen of his turrets and allies (and not him, I assume). Regardless, the passive doesn't even seem noticeable and seems completely irrelevant by mid game, especially when the adc can two-shot the turrets anyway. I don't have any ideas for what a better passive for him might be, but his current passive if very underwhelming to the point where it might as well not be there at all.
Specifically, what kind of weakness do you feel would be appropriate compensation, if he had less of a hard time with his current hard counters?
Unfortunately, Heimer is pretty balanced that weakening one aspect of him to strengthen another would probably take away his potential ability to deal with counters. We know how Riot works. Look at Cassiopeia, for example. They moved all of her damage into her E and centralized her damage around that one attack. If they did that to Heimer they would either make him completely invested in his turrets, or make his turrets so useless that there's no point to them.
Like, I mentioned earlier that he can be more of a poke lane opponent rather than playing in a turret safe zone. Theoretically, if Heimer is against, say, Annie, who can knock out his turrets from a safe range, then instead of maxing his Q first for his turrets he could max W first for a stronger lane poke (or E if you're really good at landing that broken hitbox skillshot. Likewise, as support, you should max E first for the CC) since his turrets would be, in this case, less effective and free gold to the enemy. Except that would be a more aggressive high-risk high-reward playstyle and more exhaustive on his mana. But, at the cost of his turret power, Heimer is building more for bursting with W and E.
I guess what I'm trying to say is that Heimer is one of those champions who can still be flexible in his playstyle depending on the situation, which is a good thing and Riot should stop specializing every single champion. So nerfing one aspect of him to buff another would make him more biased to one type of playstyle over the other.
I just think he could be built with sharper intent, and have his less interesting strengths exchanged for something that supported his more interesting and more appreciated playstyles better. While I don't see him very often anymore and he's probably is pretty balanced, It'd be interesting to see if his kit could be a more engaging for him and his opponents.
