I don't know if you will need to get your thyroid biopsied or not.
Most thyroid biopsies are fine needle biopsies of nodules that are suspicious for cancer. Since your goiter is described as heterogeneous echotexture with no discrete nodules identified, it seems unusual that they would recommend a biopsy. (I'm not a medical doctor, just a person who has had thyroid ultrasounds and suspicious nodules, so I could be wrong. I had a multinodular goiter that was full of nodules about 1 cm in diameter and larger - I also had Hashimoto's and a heterogeneous echotexture, and because some nodules had more suspicious characteristics, I needed a biopsy.)
A heterogeneous echotexture is often found in people with Hashimoto's and is not considered dangerous. (If there were nodules noted, they may need to be checked, but often even those are not biopsied until they meet specific size guidelines.)
I'm so sorry you are having chest pain and discomfort swallowing, breathing, and clearing your throat. If the cause of that is the thyroid, and if it is obstructing your airway, they may recommend surgery because it's important to be able to breathe.
I won't comment on whether the hypothyroidism might be causing the chest pain and fast heart beat (that's often a symptom of hyperthyroidism or too much thyroid hormone, but hypothyroidism can also cause those symptoms). I had anxiety frequently when I had hypothyroidism and would have a rapid heart beat. Hopefully they will investigate the chest pain and rapid heart rate further to make sure something else is not going on. (If your goiter is blocking airflow, this could also be causing chest pain and a rapid heart rate.)
You asked if your thyroid is REALLY large. I think normal thyroid lobe size in adults is around 4-6cm x 1.3-1.8cm x 1.3-1.8 cm. So it looks like some of those dimensions are increased by about 1 cm, maybe a little more. For comparison, here are the dimensions from my multinodular goiter: right 6cm x 2.4cm x 2.3 cm, left 5.8 cm x 3.1cm x 3.1cm, so I wouldn't say yours is "huge" in terms of a goiter, but it is enlarged compared to a normal thyroid, like mine was.
Your doctor is going to want to make sure your airway is unobstructed. If your thyroid is impeding airflow, something may need to be done, regardless of the size of your thyroid. I had my thyroid removed (some of the 1cm and larger nodules were cancerous) and because of Hashimoto's, I had a bunch of enlarged lymph nodes removed from my neck as well (14 in total - 8 pulled out in a giant lymph node clump). The lymph nodes weren't cancerous, but they were a sign that my Hashimoto's was running amok and causing lots of inflammation in general in my neck. So some symptoms you may be experiencing like trouble swallowing may be from inflammation caused by Hashimoto's, but not necessarily just a swollen thyroid.
I don't know if any of this was helpful - the main point is it doesn't sound like your goiter is suspicious for cancer. But definitely make sure it is not preventing you from breathing. I'm so sorry you have to deal with this and hope you feel better soon.