I'm concerned the H1 and other headings don't look as good as the regular text.
Are headings that important for SEO?
Also, does the use of heading look as good as without?
Yes headings are important for SEO! There is a lot of information out there about how they affect SEO and how to use them. Check out my screenshot though - The product name is actually coded as the h1 so you are ok to not include a second and if you do I would suggest h2 and something different than the product name.
It's not ideal to skip. There are reason's to keep it in order such as screen readers and accesibility, but it isn't a make it or break it type of thing for SEO it would just be best practice to have an h2 before h3. If you are wanting to use h3 instead of h2 because of size that can be done within your themes CSS.