Toilet water valve tightened to OFF position but water keeps filling up tank
It's fairly common for gate valves to fail, causing them to not open or close properly. In this case you can either repair them (most of the time), or replace them. If you're replacing them, I'd recommend installing a ball valve instead.

I'm not sure what could have caused the fill valve in the toilet to break off. Without a photo or two of the broken valve, it's difficult to even speculate what could have caused it.

As for the trickling hot water... It's not clear if you mean it always trickles out, or trickles out when you open the hot water valve.  If you mean that it trickles out when a hot valve is open, that's completely normal.  If the main water supply is turned off, there's no pressure to push the hot water out of the tank. Remember, the hot water tank is supplied by the cold water supply line.

Without being able to come on site and investigate, it's difficult to say for sure what's going on. Obviously, you'll want to replace the broken fill valve. You'll also want to repair, or replace the shutoff valve for the toilet. If you don't have the tools or knowledge to do so, you might want to have these things handled by a plumber.  

You also might want to investigate whether or not there's a pressure regulator installed on the main water supply line. If there's not, you might want to consider installing one. If there is, you'll want to make sure that it's set, and working properly.