What are the reference ranges for the Free T4 and Testosterone? Reference ranges vary from lab to lab and have to come from your own report.
You already have hypoothyroidism, so yes, you should get the antibody tests done to see if Hashimoto's is the cause of your hypothyroidism or if something else is causing it.
While the thyroid and reproductive system are both part of the endocrine system and the hormones all work in sync, just having the thyroid hormones, especially, such a low dose as 25 mcg, is unlikely to increase your testosterone level.
You should talk to your doctor about testosterone replacement therapy, as well as other pituitary tests to make sure there's not something else amiss.
Are you seeing an endocrinologist or a primary care physician? You might want to get a referral to an endo...
Your doctor should also have ordered Free T3, along with the Free T4 and TSH... Free T3 is the thyroid hormone that individual cells in the body actually use, while FT4 is more or less a "storage hormone" and must be converted to Free T3 prior to use... It's possible that your body isn't doing the conversion adequately.
Your FT4 is at 58% of its range, which is higher than the recommended 50% of range... so your thyroid seems to be producing adequately. You need to find out why your pituitary gland thinks you need more thyroid hormones, when there appears to be plenty... the FT3 test will help determine if there's enough thyroid hormones or if adequate conversion is taking place.
Levothyroxine will replace missing thyroid hormones, but it will not prevent further damage to the thyroid.