The situation is dangerous as you do not have a improperly functioning Temperature & Pressure relief valve as now installed. This safety valve is meant to bleed excessive pressure (150PSI) and temperature (210 degrees) from the water heater. It's malfunction could lead to a water heater explosion (GOOGLE).
The expansion tank is needed (on cold side) to absorb pressure fluctuations (as evidenced by the leaking T&P Valve) from either external or internal pressure surges. IMO, a PRV (pressure reducing valve) should also be installed to limit the supply pressure to no more than 60PSI.
What this guy has done is dangerous and needs to be corrected ASAP.
Thanks. I am pretty handy myself, and I looked up some videos regarding replacing the temperature and Pressure relief valve. That's something I can do on my own to mitigate the immediate concerns.
For the other items, I would like to hire a plumber. Can you summarize the items you think I need in your opinion?
Thanks again.