Lawn mower will not start after running out of gas.

Yes, clean the air filter if it's a foam , washable fliter. If it's not the washable kind , replace it. Also replace the gas filter if there's one on the engine. Buy a can or bottle of carburetor and fuel system cleaner and put it in the gas tank to clean the carburetor.

If you did not try to mow the lawn, but shet the motor off before there was a loud KA BOOM.
If not, it is time to consider a new motor.

Otherwise just drain the crankcase like you would to change the oil.
If you have a friend that could help tilt the mower so almost all the gas is drained.
Now, put oil in the motor, 40 wt because there will still be a little gas left in the motor and when you mix gas and oil the gas is absorbed by the oil and the oil will thin out.
DO NOT! start it up and rev up the motor. You want to let any remaining gas to be taken up by the oil. Run around your lawn a few times at a moderate speed WITHOUT the blades engaged.
After this you can mow the lawn, at a moderate throttle setting, then drain the oil and fill it with the oil your owner's manual says to use.

Or just put around your yard for 30 - 40 minutes, then change the oil.

The above will only apply if you didn't mow until you had flames coming out of the exhaust and banging, knock ing noises were coming from the motor.

Low compression ( head gasket ) would account for bad restart.
The 'Hitting a brick' sounds like your carburetor Seat is not working properly and gas is getting past the carburetor ( hence the wet Primer) and getting down into the cylinder. That Hitting a Brick feel was "hydro Lock" TOO MUCH GAS IN THE CYLINDER. See if your Oil is way full and smells like gas. This is being seen everywhere since our idiot politicians created ETHANOL gas

sounds like you may have a bad coil. you getting gas but no spark if its sputtering can cutting out or it may just need re gapped

Usually this is a sign of oil being in the ignition chamber. Check the oil level, if too high drain and refill to proper level. Also if you have hills or slopes in your yard make sure that you are mowing up and down not across. If you must mow across the slope make sure to do it with the oil to the downward side. Caution if you do this too often the engine will be deprived of oil all together and seize up.

Sounds like you checked everything but the gas can.

check fuel in the tank
check for blocked filter in the tank
check for full fuel flow to the carby
overhaul the carby

I typically do 6 things in this situation:

1. Verify that the gas tank is full and closed
2. Verify that the oil is filled to the "full line"
3. Remove and inspect air filter clean/replace if necessary
4. Remove and inspect spark plug clean/replace as neccesary
5. Apply warm air to engine block via hair dryer at the spark plug for 5 minutes and attempt to restart
6. If that doesn't work allow the mower to "rest" for 30 minutes and repeat step 5.

Check for water in gas tank and filter.

Carb. float must be stuck open. First if you have no fuel in the tank it must all be in the crankcase.Drain the oil and refill and also check the carb as the float has to be sticking open. Hope this will help you. Donnyb60