The smell of sewage emanates from the sink when I run the hot water.. and when I flush the toilet in the basement the area near the hot water heater makes a loud gurgling sound... what could be causing this...

You have several unrelated issues here. Starting with the smelly hot water.

It sounds like some minerals have leeched into your water and have interacted with the magnesium anode rod in your water heater creating hydrogen sulfate gas. The rod is located under the large looking nut on top of your heater. First try flushing out your heater. Attach a hose on the boiler drain located at the bottom of the heater. With the pressure left on open the hose bib and let the water run until it's clear. If the problem persists, change the anode rod to aluminum. Now check the screens on your aerators. If they are silted up clean them. Remember how you took them apart. If the shower has a single lever valve pull the cartridge and check the cold water ports for trash or silt in them. I'm betting the smell is worse from the hot water. If that's so the heater is at fault.

when I flush the toilet in the basement the area near the hot water heater makes a loud gurgling sound. That sound tells you that you have a blocked vent and the noise you hear is the discharge attempting to vent through a fixture trap.
The solution would be to snake the vent that vents the noisy trap. Good luck, Tom