Could I upload another author mod, if it's not available in twitch or not available for needed game version?
If I leave a link to official mod page and author in the description, certainly.
P.S.: I saw some forks of another mods that are not available for newer versions of minecraft, so can I do the same?
It depends on the license for the mod. Some mods will be licensed in a way that would allow this, most are not.