(1) I don't actually know what my POH says to do on tyre burst - I wonder how long it would take me to work out that that was the problem anwyay? Having diagnosed a tyre burst I'd have trouble thinking of anything better to do than stopping, braking on the non-burst side as seemed helpful, and shutting down the fuel and engine if convenient.
No need to worry about closing the one and only runway, unless you are in a very isolated location - everybody else in the circuit must have sufficient fuel and sufficient means to navigate to an alternate, so no problem there. (I know there are some people here who don't take the legal requirement to carry an up to date map seriously ... but in an area where you can see three alternates from the circuit, or if you fly the local area every day and know every local airfield, the lack of a current map wouldn't bother me.)
(2) Surely there can (almost?) never be a reason not go to around if you think it might be helpful? Circuit procedures are designed so that the joining aircraft can see what the one on the runway is doing, including go-arounds.