Is it normal to have an enlarged thyroid but no other symptoms?

Yes, its possible. As a matter of fact, most goiter presentations here in the US occur in the absence of thyroid symptoms! The enlargement usually is due to multi-nodular goiter (MNG), early stages of Hashimoto's or Graves' disease (the latter is the most likely to present with symptoms).
Yes. An enlarged thyroid should be investigated with thyroid lab testings +/- thyroid ultrasound. If labs and imaging studies are completely normal, then all you have is just an enlarged thyroid which might be your anatomical variant.