I would tend to use slog it out more for two people/companies/teams that are in competition. For example: "Apple and Samsung are slogging it out in the smartphone market". "Chelsea and Arsenal slogged it out for ninety minutes" If we want to use slog in a way that is similar to fight through, I would say something like this: "It was a really hard slog"

I would never say slog through something. Fight it out would come more naturally to me, but more than that, i would probably say "work through something"

It may vary slightly between countries which use English for mother tongue. In New Zealand we'd probably add a 'just' in the sentence and combine 'I will' (English speakers very rarely say "I will") so it would become: "I'll just slog it out." or "I'll just fight through it."

Hi. Both are correct but most people would probably say "I will fight through" rather than the first. Hope that helps! :)