in short. all Mining pools run a full node?. or some of pools are not?.
All mining pools should really run a full node. I'm sure there are shortcuts but it might make them a bit slower.
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?
Something tells me they might not get accepted if they don't run a full node. Or they'd get DDoS a rediculous amount if they tried to get away without running one.
What is forgotton is that miners should also be the main node runners and they should be able to distribute the blockchain. And why wouldn't you want to distribute it if you are earning money from it already and a greater seeding of the blockchain might make more people get into it. I'm sure F2Pool, Antpool and ViaBTC use some sort of script that goes through the mempool anyway to gather data on it with a base of a full node.