Ut shows movement while ute shows location.

We do have the expression ute ur garderoben like English has "out of the wardrobe", so saying just ute is not unheard of, but it's also not common. Instead, we'd say har kommit ut (has come out).

There's also outad, a direct - and colloquial - loan of the English "outed". It has the same meaning as its English counterpart, so it's not restricted to sexuality.

Yes, that works. Borta means he's gone, ute means he's out.

Looks like that would be "Han är borta" instead.