Phenyl rings and iodine go together like peanut butter and jelly, for reasons outside the scope of the question. For thyroid hormones though, an enzyme is used called thyroperoxidase which handles both the management of iodine's oxidation state, and its addition to tyrosine residues.

That's another good question, which again deals with unambiguous signaling. Remember that molecules have not only size in space, but also a shape and a pattern of electron density. So, yes, you could make a protein as big or bigger than T3, but will it fit into the necessary active site the same way? And will it electronically interact with the active site the same way?

For peptides, the bond angles of the main chain are given by something called a Ramachandran Plot, which works out the most likely shapes a peptide bond can form in space. In this case, maybe you could have a protein with a pair of tyrosines that could meet up and maybe look kind of like T3? Well, the necessary bond angles make that unlikely, so such a shape would not persist in space for enough time to matter. Even if it did, you'd have the rest of the protein hanging out at weird angles, preventing a good fit with the intended receptor.

Finally, even if all that worked, there's still some intentional interaction with the iodine substituents necessary to induce a tight fit. You could get a couple tyrosines in there to fake the rough shape, but how are you going to get protein parts in there to fake the iodine interactions? There's no way. (A) nothing else is even kind of shaped like those iodine substituents, and (B) even if there were, you couldn't stick them into that binding site without popping the whole protein out. Thus, the iodine groups help ensure that only T3 will reliably fit into the intended receptor site.

Others have made good points about halogen availability in marine life, but I think it's also good to look at it from a biochem perspective. T3 and T4 use phenol rings with iodine substituents sticking off them. The size of these groups will be pretty fuggen big compared to organic groups with similar functions. This makes sense, because thyroid hormones have a direct effect on the overall pace of metabolism across the whole organism. You want that lock to turn with a big key, so that it won't easily get accidentally triggered. Put another way: the harder it is to accidentally trigger thyroid hormone receptors, the better an organism will be at autoregulating its metabolism.