"My cousin and I have a watermelon."
Translation:Ο ξάδελφος μου και εγώ έχουμε ένα καρπούζι.
That's not the reason, though, it's not about mixing up the plurals or not.
The same argument could be used in the case of "Είμαι μαθητής.", where "Είμαι ένας μαθητής." sounds kind of robotic and the article is sometimes only included to make it a bit easier for learners (1-1 matching). It doesn't have anything to do with μαθητής-μαθητές.
"Έχει καφέ μάτια" or "Έχει δύο καφέ μάτια;"? The former, for sure. It's about being practical.