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I purchased two RES about 2 weeks ago.
The tank is 10 gallons with UVB light and water heater, with a rock for basking.

One of the turtle is very active the first few days I bought it. However, the last few days, I often see him basking extensively on the rock with limited movement. Is it normal? He still eats the food pellet I drop in the tank.

Unfortunately for the other RES, it has just died. I suspected it was through drowning. I just found it up-side down being trapped by the rock. However, I also noticed there's a small white spot on one of it's front leg. Is it possible that it is an infection? If so, will it spread to the other turtle?

I am quite worried about the state of the remaining turtle. They seemed to be getting along real well as I find them sleeping together when I wake up.

Hm... I'd suggest keeping the water temperature at around 80°F and basking
temperature at around 90°F. You need to use a thermometer, you can't just rely
on your fingers. That is to bump up the turtle's metabolism to fight germs and
whatnots.

Eating is a good sign. Keep watching your turtle if it gets more lethargic or
developing swollen eyes, mucus coming out from nose and mouth, inability to
submerge, swimming lopsided, etc. Those are signs for Repiratory Infection (RI),
quite common in hatchlings but can be deadly. If it is RI, you need a vet's help.

Keep us posted and good luck!