Will adjusting my thermostat on my hot water heater to hot make the hot water last longer?

For showers that long you should get a "Whole house tankless water heater". It will heat on demand until you turn off the hot water.

A fifty gallon water heater has 40 gallons of hot water. If you are using a water saving shower head of 2.5 gallons of water per minute that will give you 16 minutes of hot water. There's no way it can provide a 30 minute shower. Although raising the thermostat will increase the temperature it greatly increases the risk of scalds and burns.

Yes if you mix hot and cold water as you do in a shower. You'll need less hot water to raise the temperature of the cold water to a desired level. It takes less 140 deg water to raise cold water to 90 deg. However, it will cost you more money to keep that water that hot.

No if you use hot water only. If you have a 50 gal water heater, you can only get 50 gal of pure hot water out of it. Actually, less than 50 gal because cold water coming in tends to moderate the volume of available hot water.

Not much. You most likely need to drain the hot water heater and try and flush the sediment out of it.

Over the years sediment and calcium deposits from the water build up in the bottom of the hot water heater decreasing the amount of hot water you have available. Just by hooking up a water hose to the drain on the bottom of the water heater and open it wide open should flush it good.

If nothing comes out of the heater then it is too late and will need replaced soon.

It will definitely last longer for showers because you wont be turning the hot water on so much. Insulate the tank for better efficency as the fuel costs to maintain the hotter water in your tank will go up.

Raise it a little at a time until you get the desired temperature

No, but taking a more reasonable length shower will.

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