The vinyl tile can be removed with a tool from the rental place. It has an oscillating blade on the front of it that is just the right angle to scrape the floor clean (its electric). Thing is, it can't get the old adhesive off the floor, which could be a problem. BTW I hate vinyl tiles too.
If you have hardwood floors under the plywood, I would go for the remove plywood and refinish. Once you get all the nails out and strip the old varnish off, the floor is not too hard to refinish. It can be sanded with a rental machine that Lowe's has. It is a square, orbital sander that won't dig into the floor but gives a nice smooth finish (drum sanders dig in before you can say 'don't do that').
Lamanite floors are a good attempt at copying the hardwood floors, but if you already have hardwood, it will look the best.