Is TSH the only test you had done? What about Free T4 and Free T3? Those are the actual thyroid hormones, whereas TSH is a pituitary hormone that stimulates the thyroid to produce hormones. TSH, itself, causes no symptoms; it's the presence or lack of thyroid hormones (Free T4 and Free T3) that cause the symptoms.
Do you know if your mother or sister have autoimmune conditions that caused their over active and/or under active thyroid? Graves Disease most commonly associated with over active thyroid (hyperthyroidism) and Hashimoto's is most commonly associated with hypothyroidism, though many have periods of hyperthyroidism in beginning stages.
You should ask your doctor to test for thyroid antibodies to determine if you have Hashimoto's. If you do, it's a lifelong condition and the answer to your question would then be yes - your TSH would, most likely, continue to rise, because your thyroid output will decline as the antibodies destroy its ability to produce hormones.
The answer to your question about symptoms is that, while TSH doesn't cause the symptoms, it's possible that your Free T4 and Free T3 are low enough that you could very well have symptoms of hypothyroidism, with a TSH of 4.51. Most people would have adequate hormone levels to maintain a TSH of around 1.0 -2.0...
What's the reference range for the Free T4? Ranges vary from lab to lab and have to come from your own report.
Autoimmune conditions tend to run in families, though not each member of the family will get the same condition... For instance, I have Hashimoto's and Pernicious Anemia, my son has Type I Diabetes and my daughter has Lupus. When one has one autoimmune, the chances of getting another are greater... You should ask your doctor test you for thyroid antibodies to determine if you have Hashimoto's.
The tests you need are Thyroid Peroxidase Antibodies (TPOab) and Thyroglobulin Antibodies (TgAb)... You need them both, since some of us have one or the other and some have them both.
Hashimoto's is the most common cause of hypothyroidism (under active thyroid) in the developed world. Irregular menstrual cycles are one symptom of hypothyroidism, as is weight gain/inability to lose.
You should also ask for a thyroid ultrasound... your "sore throat" could be caused by a swollen/inflamed thyroid - also known as a goiter. An ultrasound will also show if there are nodules on your thyroid. Nodules are, typically, nothing to worry about, but it's important to know if we have them.
You sister should also be tested for thyroid antibodies to determine if she has Hashimoto's. There is a connection between kidney function and thyroid function.