in short. all Mining pools run a full node?. or some of pools are not?.
If you want to run a pool, you need access to a node... So yes, they are either running a node, or they are paying somebody to run a node they can query.
The pool needs at very least the header of the previous block, and if they don't want to mine blocks with only the coinbase transaction, they also need information about the broadcasted transactions... And if they want to include these transactions, they need to update their utxo set, so they need to parse the complete blockchain... In other words: they need to run a node
I think what OP is asking is whether mining pool operators use a full or a pruned node.
In general it seems to make sense for mining pools to run a full node, since if you already harness millions of dollars worth of computational power there's no sense on skimping when it comes to running a node, amirite?
But from a technical perspective it seems like they could run a pruned node just as well, since they only care about current transactions. Or would a mining pool be prevented from publishing blocks if they are not connected to the network as a full node?