Toilet Tank Gaskets

"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. 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.

"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.