You can find super cheap wireless keyboards at Target for like $8-$12. It's a nice investment, if only to involve yourself with other players. ("Good job everyone, thanks for the group!") Until then, I'd recommend making a macro for the beginning of the dungeon; "Hello everyone, glad to be here. Just so you know, I don't have a keyboard, but I'm still listening!"

Secondly, unfortunately, FFXIV's entire armory system is based on roles defined by your primary weapon. So yeah, if you're going to want to hold a sword, you're just going to have to be a Gladiator. Maybe later Square will develop more horizontally, and make equipment that will make a DPS Gladiator viable, but for now, they're just trying to get a full-sized game out for their players.

I think you'd enjoy Lancer. Give that a try.