Yes, I recommend a slight over-exposure. You can set whole stops using your shutter (i.e. if the meter recommends 1/500th, set it to 1/250th), but your lens should be able to set in-between stops (i.e. you should be able to set the lens aperture halfway between f8 and f11). Given how cheap these films are, and the fact that you don't know how badly affected they are from their storage conditions, I would shoot a couple of test rolls, bracketing exposures (take one frame at the metered exposure, another 1/2 stop over, another 1 stop over, of the exact same scene) and then get it processed and see what turns out best. If these were negative films I would recommend bracketing in larger increments (1 stop, 2 stops over), but slide film is much more sensitive to over-exposure in a bad way.
I would store the tri-x in the fridge only if you are not planning on using it soon (in the next few months). Given that you said these films have not been refrigerated for what may have been an extended period of time already, another few days/weeks won't hurt it any (unless those days/weeks are going to be spent in the boot of an un-airconditioned car parked in the sun). If your fridge is freezing your eggs, then A: it's too cold in general, and B: if you store film in it, to avoid freezer burn/frost damage, you should seal the film inside a ziploc bag with the air evacuated as much as possible, and then that bag inside another ziploc bag. This will help prevent condensation from forming when you defrost the film. You will need to take the bagged film out of the refrigeration several hours before you plan to use it and allow the sealed bags to come to room temperature before opening.
Another note: given that you are still learning, my recommendation would be to buy fresh film and learn with that so that you see what your results SHOULD look like. Don't be economical to the point of compromising your knowledge and results just for the sake of saving a few pence/cents/rupees.