Different Pressure in Parts of the House
You possibly have galvanized steel pipes which when replaced with PEX or copper would fix your low water pressure. I'm not sure why galvanized steel was ever used for water pipes, but you can use a magnet to tell if they're steel or not. Other than being a different color than steel, copper isn't magnetic and steel is magnetic. 

Your T&amp;P valve on your water heater will leak for 3 reasons. 1 if the pressure is too high, 2 if the temperature is too high, or 3 the valve is failing. 

Reason 1, your pressure too high, can be corrected by adding an expansion tank to your hot water line. Water expands when heated, and it needs somewhere to go or it'll build up a pressure much higher than your normal water pressure. 

Reason 2, your temperature is too high, can be corrected by turning down your water heater temperature. The US Department of Energy recommends as low as 120F, 140F is the most common, and it really shouldn't ever be set higher than 160F because the scalding/burn risk is extremely high at that temperature. 

Reason 3, your T&amp;P valve failed, is unlikely since it is new. Typically they fail where they never seal or never open. So it leaking only when the heater turns on makes it more likely to be reasons 1 and/or 2.