Identify a card game where you need to play a higher card or pickup the discard pile?
I believe you are describing Shithead, or one of its MANY variants.
I played this at school in 1990's and when I went to University people from different places had the numbers special powers have slightly different names.  But what I played broadly matches your description.  As stated in the wikipedia link the game does have many regional variations. I am describing this from UK.
In the version of Shithead I played there were 3 cards face down to each player.  3 face up on those cards and players could then swap those face up cards with ones in their hand.  In the version I played all cards were dealt out and sometimes we played with multiple decks in a large group.
In the version I played 2's reset the deck.  7's were transparent and could be played on anything and 10s 'burnt' the stack removing them from the game.
Once a player had emptied their hand they would play with the 3 face up cards and then finally the 3 face down cards were played blind on a players turn.