“Pair” is a noun with two forms: pair (singular) and pairs (plural).
And there is a determiner “a”.
There “is” a pair of socks, because we are talking about the pair, not about the socks (maybe this is what is confusing you).
Your cousin was wrong.
On the other way, I do not understand the sentence. Perhaps, it was like this:
There --- a pair of socks and two shirts on the table.
In these case, you would say:
There "are" a pair of socks and two shirts on the table.
A pair of socks + two shirts = ARE.
José Manuel Rosón Bravo