In PoW system, a node can trustlessly check for validity of the whole chain just by checking the hash of every block correspond to the prev hash in the descendant block header. As far as I know in the PoS validators are selected randomly, and the probability is proportion to the amount of coin they have staked. So if it is a random process, I think it's not reproducible in the exact same situation, opposite of PoW. So can someone validate the whole chain from genesis to the tip, without needing to trust anyone?
Based upon system requirements, every PoS blockchain network is constituted of more than a single validator. This validator is selected randomly to propose and validate a block, and a single block is validated by more than one validators, and a block is only finalized if at least two-third of the validators agreed on the state of the block at a particular checkpoint. Hence, it's extremely unlikely someone can validate the whole chain from genesis to the tip, without needing to trust anyone else.
If you mean to check if the current state of the blockchain is correct - then most certainly yes. As a matter of fact, when you decide to run a node, you need to download the whole blockchain. In that process, all of the transactions recorded on the chain will be run also locally on your machine and you should be able to reach the same state as everyone else - same state in a sense of all the Data Tries. That process usually takes few days (70+ hours)
If you mean to validate all of the transactions as you go during every epoch - then no. You will be put in certain slot committee every epoch, meaning that every epoch you can validate (or attest to) at most one block.