Could just be a runny nose from the teeth coming in and cough from post nasal drip.
My first kid had a really hard time with his 1st set for sure! When his first one was coming in, he as just screaming and screaming for 1/2 hour minimum, and I had no idea what was happening. Figured it out a couple days later. We just used Ibuprofen a lot and those Hyland's Teething Tablets.
My second little guy is currently getting in 6 teeth at once (late teether). 2 molars, 3 lateral incisors and a cuspid. All on the top except for 2 of the incisors! He's taking it pretty well. He's cranky, but teething tablets seem to be enough.