We recently had a friend replace the faucet fixture in our main bathroom and it has been working normally. Last week my wife clogged up the drain in that sink with her hair cuttings and I shut off both hot and cold water intake valves located under the sink in order to prevent the kids from filling the basin further. After using drain cleaner to unclog the drain I turned both hot and cold water back on but now the hot works normally but the cold is only a trickle. If I turn off the cold water intake and open the faucet it still trickles out even though it should be off. The rest of the house works normally for hot and cold. Is the problem at the new cold faucet handle or a clog in the cold intake tubing or at the valve?

What has happened is that by turning off and on the cold water angle stop you broke loose some mineral build up in the supply line and now it's clogging the cold water in let port on your cartridge.
Try this. Shut the water off and pull the cartridge, Check the inlet port for trash and clean. Now place a pan over the open valve and turn the water back on for a minute to flush out the supply. You should have a strong stream when doing so. Reassemble and test. If that works do it on all affected taps, good luck, Tom