yes, thermostat..
honestly, can't say I've ever seen a water pump fail due to no fluid in it.. I wouldn't do it till it leaks from the weep hole, or squeals...
if you have signs of leaking in the rad, I think you answered your own question.. test with water, replace the rad more than likely.. I would start with a heavy look at it..
if the rad core isn't leaking into the cab floor, and you can run a garden hose in one hose and out the other, it's fine...
another thing here..
IF you determine it's the rad.. do yourself a HUGE favor before you pull the old one.... first find your block drains on either side right above the oilpan lip.. they should be 2 pipe plugs with a 9/16 head more than likely, on ds and pass side...
pull them out and drain the block.. they may be blocked with crud and need a poke with a paper clip or similar.... drain.. replug.. refill with water.. run to temp, shutdown.. drain again (careful of hot fluid).. replug..
fill rad and add a couple cans of prestone flush... run to temp, shut down.. drain... refill with regular water, run to temp, shut down... replug....
NOW, replace your rad.... that does everything it can to clean the system PRIOR to putting the fresh rad in... go from there....