"I installed a new fill valve and my new valve comes on (and then goes off) for a few seconds about once per hour?"
This could be caused by many things. The most common is with a Fluidmaster 400. This is a great valve but be sure to install the 1/4" tube per their instructions. Use the clip that comes with it and do not just stick it into the overflow tube or it will go on and off every so often by itself. If that isn't it, take some food coloring and place it in your tank. Wait a few hours and if some of the color has gone into the bowl then you probably need a new flapper.

"Why does my toilet continue to run after flushing?"
If, after flushing, you find that the toilet continues to run, and you have discovered that by jiggling the handle it will stop; then the problem may be that the chain controlling the flapper is too long. When installing your flapper, you want the flapper chain's S-hook in the hole closest to the handle with very little slack in the chain itself. By doing so, when the toilet is flushed, the chain will fall behind the flapper where it won't snag and cause problems such as keeping the flapper from closing properly. Check to make sure that the flapper has enough clearance to close, i.e. not hitting the float ball when the water level drops. You also want to check to see that the toilet's flush handle (aka tank lever) is installed securely so it can't rotate too high, hitting the tank's lid or other parts and sticking.

"What's the best way to replace my toilet flapper?"
Generally, with most toilets: turn off the water supply and flush the toilet. Reach in and unhook the ears of the flapper and unhook the chain from the trip lever. Install an identical flapper to the original that came from the factory. Do expect to get your hands dirty from the old flapper. Simply reinstall the new one in reverse order. Note that should you have very old brass pipes inside of your toilet, be careful not to be rough on them. They can easily break and end up leaking. After you have replaced your flapper, and the toilet tank bowl has refilled, the water fill valve should not leak (be going on and off in cycles). If it does, then we recommend putting some food coloring into your tank. This can help diagnose the problem. The colored water should not be going into the bowl (unless you flush the toilet of course). If the food color does go into the bowl, then possibly the flapper is not the correct one, the surface where the toilet flapper sits, has eroded (feel below where the flapper touched the toilet and see if you can feel erosion/grove) or you may only need to add some slack to the chain.

"How long 'should' the parts inside of my toilet last?"
That depends on a number of variables and which parts. Water quality is one of the major factors. If your water source is heavily chlorinated then many parts won't last long. Or if you have a lot of sand or grit or have a low pH or "aggressive" water source then parts simply won't last as long as the 'average' length of time. Also the quality of the parts matters. The replaceable parts such as flappers and washers/seals generally will last at least 4 to 5 years on "average." If you use a bowl cleaner some flappers won't even last one year. If your flapper has black "goo" on it then it is either due to age or quality of your water and if it feels soft then it's best to change it.