Do we need to use y here? Is it similar to saying "I said you'd be OK" vs "I said that you'd be OK"?
You don't need it: Dwedais i baset ti'n iawn is fine. Formal registers of the language include the y "that" but you don't need it in colloquial Welsh.
Why is there a 'y' here in front of 'baset'? I was expecting 'that' to be 'for' in this construction.
It's an optional word for "that" before the conditional tense ("would, could, should..."). You can leave it out: Dwedais i baset ti'n iawn.
That should have read 'fod' not 'for' - it's this damned autocorrect
Why the 'y' in front of 'baset'?