The top of the overflow tube should be at the fill line cast in or marked on the back (inside) wall of the tank - which should usually be below the handle opening, but some toilets use a lower handle with a compression gasket to prevent leakage - check for loose handle mounting bolt or nut. A toilet or tank tilted substantially forward could of course also cause leakage even if the tube is at the correct height.
Trimming the tube would solve the water coming out the handle hole, but would not solve what sounds like your primary problem - the fill valve not shutting off every time - it would just run and run into the bowl until it shuts off. Overflow tube is unlikely the issue, because presumably the toilet worked fine before now and the tube presumably has not changed in that time. Check the linkage and fill valve for proper, free operation, and that the float is not hanging up on something, laden with water (waterlogged so not floating properly) or that the slide bar it moves on (if modern fill valve float that rises vertically) is not all crudded up with slime and hanging it up. Since this sound like an intermittent issue, that or a leaking fill valve sound like the likely cause. Or the fill valve might be leaking some, causing water to rise gradually after the float has risen to shut off the valve. Lots of how-to articles and videos on the web on adjusting float setting and inlet valve for different
Shut off valve to toilet - they commonly leak around the stem slightly when used because they sit for decades without use. Will commonly, if only a VERY minor drip, block off OK after a day or two as the lime builds up. Or, holding the vavlve firmly to avoid putting stress on the pipe, carefulyy tighten the cap nut (the decorative nut right below the valve handle, which the valve stem goes through) clockwise about 1/8 turn to compress the washer there a bit more to stop the leak, assuming it is around the valve stem.
Sink and shower drips - sounds like they might need new washers inside the faucets, at the point where the valve closes, so they are not totally shutting off the flow. Plumber can fix 2-3 of these typically for one basic service charge of $75-150 typically, assuming traditional handle for each faucet (hot and cold) or a Delta or Peerless single-handle type faucet. If a fancier unibody shower valve can run $50-100 more for a cartridge rerplacement, if needed.
Water heater overheating - plumber should look at ASAP, as that is potentially dangerous. Could be boiling due to failure to drain bottom of tank yearly so bottom sediemtn is causing boiling, could be a thermostat failure - or if "blowing down" out the relief valve might just be a faulty valve releasing too easily. Costs depends on cause, so could be as little as $50-75 additional (at same time as service call for leaking faucets) for a new relief valve or to drain the tank to hundreds of $ depending on cause.
One other possibility, with several leaking valves/faucets and maybe water heater relief valve "blowing down", is that your household pressure has for some reason become too high and needs a pressure regulator adjusted - or one installed if you don't have one. Plumber (or you) can easily check this with a screw-on pressure gauge attached to a hose bib or drain valve - just NOT on a hydronic heating system - those run a much lower pressure than the typical 30-60 psi household pressure.
