I'm going to try to be succinct about the gluten/thyroid connection.
Gluten causes leaky gut, which then leads to excessive stresses on the immune system and autoimmune conditions with time.
The excessive stresses on the immune system wear down the adrenals, and the adrenal function is heavily linked with thyroid function. After progressing past that, we get into autoimmune disorders, of which the thyroid is a big participant in.
As for O-3s, without enough of them, or the right ratios, all communication within the body is compromised. This is why a lack of fish oil can cause so many different (seemingly unrelated) problems in the body, and appear like the panacea to "fix" so many problems. The thyroid is just one of them.
As I mentioned before, men don't tend to have as much of an issue with iodine. You may or may not have an issue--although I'd place my bet that you would be at least slightly low. The first issue is always making sure that autoimmune thyroid conditions are not present, both through history and previous/current labs. An autoimmune condition simply warrants a different approach than normal.
There way I suggest pursuing this issue is the 24-hour "loading" iodine sufficiency test. You take 50mg of iodine/potassium iodide (Iodoral) and then collect your urine for 24 hours. The amount of iodine you excrete of that 50mg is indicative of your body's hunger/need for iodine. Sufficiency is noted as being a >90% excretion (more than 45mg comes out in the urine). Knowing that, we can estimate the time necessary to be on a daily loading dose, and follow up with another test around that time. After sufficiency is reached, the daily dose is reduced to 12.5mg. If one wants to do whole food sources of iodine, as in significant doses of sea vegetables (one kelp tablet may contain ~500mcg of iodine, hardly enough!), that can be done after sufficiency is reached. This is the approach I would suggest.
Some have simply taken the loading dose for a period of time (3-4.5 months, let's say), assumed that sufficiency was reached, and then gone to the maintenance dose.
As a male, I wouldn't "go by feel" on this issue, as you will likely not notice huge changes unless you have significant iodine deficiency (easily found out through the test, which costs less than two bottles of Iodoral).
I do consistently notice that I do not get cold when everyone else is, and that reducing bodyfat has become ridiculously easy since I've been taking the Iodoral. I also consume a decent amount of sea vegetables (I'm going to submit my seaweed salad recipe to the PMenu).