What happens if full nodes don't check for double spends or deliberately lets a double spend TX pass and then this full node relays the transaction? I am studying about Bitcoin and Blockchain tech via the Princeton crypto course, and it mentions that Full nodes do the following checks :

i)transaction is valid with the current blockchain, 

ii)whether the scripts matches whitelist, 

iii)transaction hasn't been seen before and 

iv)doesn't conflict with others its relayed (to avoid double spend). 

However, the lecturer also mentioned these checks are not enforced on a node. Lets say a full node may have a full set of of UTXO but still decides to relay a transaction that has been spent before (effectively ignoring the fact that the TX is not on the UTXO pool) , and eventually this TX is relayed around and makes it into a block. Since the miners trust these nodes to verify transactions , wouldn't the TX be put into the consensus chain ? 

I understand that the mining process (POW) and resulting block reward can only be  claimed/collected if the block ends up on the long-term consensus branch, but this doesn't safeguard the malicious node behavior as I have explained since miners don't check for double spends.
Since the miners trust these nodes to verify transactions, ... since miners don't check for double spends.


What makes you think this? Miners are also nodes; they are a subset of nodes. Miners still receive and verify all transactions, they don't just blindly trust what some other node has given them. Miners are incentivized to check transactions as including any invalid transactions means that the block will also be invalid and thus be rejected by the other nodes and miners on the network.

If a miner produces an invalid transaction (e.g. by including a double spend), the block will be considered invalid. Miners do not determine what is valid and invalid, nodes do. So when the nodes on the network receive that invalid block, they will reject it as invalid and not add it to their blockchain. Furthermore, since miners are also nodes, other miners will reject the block and thus they will not mine on top of it.

If miners mine on top of the invalid block, there will be a blockchain fork. The nodes on the network will refuse to follow a chain that contains an invalid block, even if it has more work. They only follow the valid chain with the most work, and an invalid block makes that chain invalid.