If your thyroid legion is completely fluid filled, then it is a cyst and usually benign. Cysts can also be aspirated, especially if they are large and are impeding other structures in your neck (like your trachea or esophagus) or causing pain or discomfort. Sometimes they can grow in size rapidly since they are fluid filled (unlike solid thyroid nodules which usually grow very slowly).
A fine needle biopsy on a cyst probably won't yield many cells if it is fluid filled, some thyroid nodules have solid and cystic components and those do need to be biopsied, but it doesn't sound like that is your case.
If your cyst isn't causing pain and the pathologist has described it as completely fluid filled, there is a good chance your doctor will recommend just leaving it alone and possibly doing follow-up ultrasounds (at 6 months or a year) to see if it has changed in size. Since it is only 5mm, this is pretty small and they might not recommend draining it. Even if it was a solid thyroid nodule (which still has a low probability of being cancerous), most doctors won't recommend a biopsy until the nodule is at least 1 cm (10 mm).
If your family history of thyroid disorders includes medullary thyroid cancer, I would definitely tell your doctors about that because medullary thyroid cancer is rare and often hereditary. Otherwise, I think just monitoring the cyst and seeing if it changes in size is a good approach.
I don't know percentages off the top of my head, but I know when men find nodules it is often more likely to be cancerous, like you said. A lot of factors come in to play with that. Lots of people have thyroid nodules (up to 80% of all people) and most will never know about it because there is no reason for neck ultrasound, CT, Xray, etc. A lot of time nodules are only discovered when they reach a large size that is easily felt by doctors or can be seen as bulge on the neck, and nodules larger than 2 cm are more likely to be cancer than if they are smaller than 2 cm. Since you have a cyst, and it is small, your chance of cancer is at a much lower risk than someone who walks in with a 4 cm solid nodule.
I'm sorry you have to go through this - I know finding thyroid nodules and cysts can be very scary. It sounds like you have a cyst, which is usually benign and not something you need to worry about, but monitoring it to see if it changes in size is probably a good idea.
You should have at least a fine needle biopsy with pathology by Endocrinologist before they can say it's nothing.