If battery tech continues to improve at current rates then certainly electric jetpacks will be possible and practical, if there's still demand (and we don't destroy ourselves one way or another). Without doing the math though it might be many decades.
Unlikely. The specific energy of jet fuel is more than 40 times that of batteries, and I don't see batteries improving by an order of magnitude. A jetpack would need 40 times more battery mass than fuel to produce the same thrust for the same duration. Of course, the additional weight of the batteries necessitate even more thrust...
Existing aluminum-air batteries have 1300 Wh/kg, or 4.7 MJ/kg in modern units, and it's plausible they could improve to over 20 MJ/kg. The more expensive lithium-air battery already exceeds 6 MJ/kg. Jet fuel is 43 MJ/kg. Pure aluminum is 31 MJ/kg, so to the extent that you can build "new Al-air batteries" during flight with the cathode and electrolyte from "old batteries", you can approach that limit. That sounds ridiculous but we've been doing it with hydrogen-oxygen fuel cells since 01838. Hydrogen has 118 MJ/kg, and there's no fundamental physical reason hydrogen-air fuel cells can't be epsilon above that, though current models are large and heavy.
It's true that all rechargeable batteries like Li-ion batteries are so far under 1 MJ/kg, which more than 40 times worse than jet fuel just as you say, but it's extremely plausible that Al-air could beat Li-ion by an order of magnitude, and Li-air already does, except for the best Li-ion cells. It's already most of the way there. And fuel cells can beat jet fuel by almost a factor of 3, and for low-power applications they do.
Betavoltaic cells and TEGs have even higher energy density but you can't turn them off, so you need to dissipate the power they produce when you're not using it to fly or whatever. You could build a hybrid system with a large glide ratio that uses chemical batteries for takeoff and a betavoltaic battery or TEG for long-distance cruising, but you might have to only land it on water to keep it from overheating while it was on the ground.
As you say, practicable batteries in production have specific energy under 300 Wh/kg (1 MJ/kg), 40x worse than fuel.
The other technologies you mention don't seem to be practical for now. I don't think it is reasonable to expect any breakthrough making it to market in the next decade. With specific energy doubling every 10+ yrs, electric flight might be sensible for niche applications (training flights, occasional urban transport for example to the airport) in a few years, but for anything else it'll be several decades realistically.
Hydrogen-air fuel cells and aluminum-air batteries don't need any breakthroughs to work; they're well-understood, proven technology. They're just more expensive than kerosene and, in the case of hydrogen, more voluminous.
To put it differently, the reason nobody is flying across the country today on aluminum-air battery power is not that the existing batteries don't have the range; it's that they can fly across the country more cheaply on a combustion-powered jet.
(I think that will continue to be the case even when the combustion fuel is synthesized with electrical energy.)
> If battery tech continues to improve at current rates
What do you consider current rates? They improve slowly, maybe 3-7% a year. Last I looked, batteries are so much worse in terms of specific energy than fossile fuels that we're looking at several decades before electric flight becomes practicable (except for very short distances and training flights).
