Hi Alexander,
A running shop with higher quality video would help you. As an amateur, from that particular video.... I'd say that you're a little bit underpronating.
You're running on your mid/forefoot, rather than heel striking... which is probably a good thing, but it is stressful on forefoot, and on the calves... which is in line with your comments on your soreness. Your calves will need to strengthen over time (don't over do things)... but you need good forefoot cushioning
You're running with your feet not parallel to one another (not uncommon... but not the most efficient method). That's something you might want to work on... you don't need them bang parallel, but something a little closer might be best.
I'll continue my amateur view, and suggest a show with good forefoot cushioning... and decent flexibility, to accommodate your inflexible arches. Maybe Asics Gel-Cumulus 14 - or some of its competitors. Brooks Ghost 5 perhaps..
I just put this out there, in the hope that others will add constructive criticism to what I've said. It's like personal challenge to see if I"m right!
Go to a running shop though (and let me know if they agree
) Good luck.