Saving my most promising theory for last.
There's one thing that bugs me about Akasake and how Rika helped him save his wife and daughter. Why would he believe her warning? And how is "someone fell down that stair that day" convincing enough for him? Quite simple, it isn't! Akasake didn't believe Rika, and got the experience the terrible fate that awaited him. As a reaction he travelled back in time to hurry back and save his wife and daughter. The story how "someone fell down that day" is just a means for him to explain it logically.
Now this also gives us insight in how his ability works. He can only travel backwards in time!
The next strange thing is how easily he agrees to Rika's request to stay in Hinamizawa. We've just heard that he's quite attached to his wife, yet without much struggle he cancels his own plans. The reason is simple, he does owe Rika and can quite easily stick around to see if something would really happen. If not, he simply goes back in time and continues his plans as normal (knowing that Rika will remain safe).
But his time-travelling abilities aren't something new. We also see it in Himatsubushi. How does such a young man win a game of Mahjong against Ooishi? We do know Ooishi is good at it (mostly through cheating). What likely transpired is that Akasake tried everything over and over, but could never win due to Ooishi's cheating. Learning how to cheat at Mahjong wasn't something he could easily do (even with his time travelling abilities), so instead he figured out the right moment to "knock over the wall", losing some points but stopping the cheating in its track.
Obviously it's important to note that his time travelling is something different than looping. As such the sword will likely not work to kill him. I'm quite puzzled why he went L5, since he's also (part) alien. Up until this point his alien traits only seem to come to expression in his regenerative powers and skill set (sharp shooting, instinct, etc...), but it was quite clear that he was not susceptible to the Hinamizawa syndrome. We probably need to adjust our theories and consider him "immune" to naturally contracting it, meaning going L5 was clearly induced. This also hints that the Ooishi, Akane and Kimiyoshi were likely also induced.
A final addition to this theory. I'll admit it's a bit sketchy, unlike the base theory which is practically certain. The alien origins can be traced back to Hinamizawa. He is actually a descendant from the village. This connection allowed Rika to appear in front of Akasaka during Matsuribayashi in the same way Hanyuu appears before Rika. Do note that this isn't the origin of his time-travelling powers! He is part descendant of Hinamizawa (oni/alien) and part time traveller, after all. (Though time travelling could just be a skill and not inherited)
Now we can speculate all we want whether the fact that he didn't age in five years is due to his alien descend or his time-travelling powers. But the real question is, why doesn't he resolve all problems in one loop? My own pet theory is that the time-travelling and looping do interfere. Despite operating independently, there is some cross interaction between them. The time-travelling of Akasaka overwrites the events that already took place. However, this doesn't happen instantly, but requires time itself. For anyone observing it, it would feel instant as memories are logically also overwritten. But in case someone loops, the point of looping could prevent some of these overwrites from taking place. As a result the looper will experience the world only in one way, with some, but potentially not all, rewrites done by Akasaka. This in itself creates temporal anomalies, which we finally got to see in episode 14!
At the start of the episode we see Rika awaken in shock. The calendar clearly reads 21st, yet a few moments later it's shown that it now reads 12th. During episode 14 the "hide and seek" game takes place where Rika finds the fragment of the sword. Since Rika doesn't terminate the loop in that episode, it simply progresses into episode 15. And guess who shows up?
Akasaka. This clearly shows that the temporal anomaly took place. Any time keeping device has been very reliable to pinpoint dates and times, so this definitely isn't a mistake or oversight by the animators.
The painful thing is, the calendar isn't on the 21st for no reason. We don't have a clear indication when Akasaka goes L5, but I now think that it would be the 21st. Instead of succumbing to it, his regenerative powers ensures he survives the sea of flames, regains his sense and travels back in time to prevent this from happening. But due to the interference between Rika's looping, she only experienced this one version of the events and goes on to the next loop. I'm sure Akasaka would've brought it to a happy end, but poor Rika simply wasn't destined to experience this
On one hand I quite like that Akasaka's powers are understated in the show and not used as a solution to everything, but would've loved seeing it play a slightly bigger role. Nevertheless, happy to see it in action again and this really helps solidify the theories surrounding Akasaka.
(In case it wasn't clear already, this one is a joke)