The supply can vary over time. It is often higher at night when people are no longer using water and the utility may be pumping water into the towers for the supply for the next day.
IF you have a closed system, an expansion tank should keep the house's water pressure very close to the street pressure (it will keep it from rising more than a little bit). The T&P valve is designed to open at 150psi OR if the temp exceeds 200-degrees. I doubt your WH is exceeding 200-degrees! So, the only way to know if the T&P is defective, or is actually doing its job is to monitor the water pressure for say a 24-hour period or more. At HD, you can buy a water pressure gauge that has a tattle-tale, second hand that shows the peak pressure. It's cheap, typically around $11 or so.
To make a closed system, there has to either be a check valve on your supply, or a PRV (pressure REDUCTION valve, not a regulation valve). When working properly, a PRV will prevent the INCOMING (not house pressure inside) from exceeding its setting. Once that water is IN the house's system, any expansion caused by heating could raise the pressure above what the PRV is set for - that is what the expansion tank is supposed to take care of.
If you have an existing PRV, it may be worn out. It won't hurt to install a new one somewhere in the line if you already have one. Normally, if an old one is buried somewhere, it won't affect your overall volume and water pressure coming in, but it might make noises.
It's bad practice to bury it in the wall since it may need to be adjusted or replaced eventually - they DO wear out, and it WILL need service eventually, just as will the expansion tank.
A PRV will not raise the street pressure, but it can lower it. One is required if the incoming water pressure ever exceeds 80psi (this does not mean internal pressure that may be caused by expansion in a closed system, the expansion tank, properly setup, should prevent expansion from raising the pressure more than say a pound or two).
You may be noticing the effects of the water being expanded now with the new WH because the new one may have a bigger burner, which raises the temp (and thus the expansion) faster. If there were any leaks in your old plumbing, say a faucet dripping, or a flakey toilet fill valve, they may have been able to bleed off that excess water (and therefore pressure) as fast as it was produced by heating. The new one, being faster to heat, they can no longer mask that problem.
Do not confuse pressure with volume. Both a soda straw and a fire hose could have the same pressure, but the fire hose will dispense LOTS more water than the soda straw. You may have not only a pressure issue (why the T&P is opening) but also a volume one. That could be caused by a too small supply, rusting out galvanized steel supply line(s), a valve not fully open, a crimped line, and other things.
As to costs to install, most here try to avoid that, as labor rates vary considerably all over the country, and often from one side of town to another.