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I agree with Roman and Donald that it's important for SEO to properly mark up your headings and should be fairly easy to change font size with CSS. It's also important to get your heading markup right so that folks who rely on screen readers and other accessibility devices can navigate your site.

When we changed domains, I moved several thousand posts, all at once—I don't know of any reason to throttle that. Our organic traffic is up 174% since then, so it definitely didn't hurt anything. (I am assuming that you will be setting up the appropriate redirects and Google will know that this is just a domain change.)

Assuming you are keeping your domain, what you should do is 301 redirect your old pages and your competitor's old pages to your new pages. You can't rename their domain to yours but you need to redirect it to yours. You will also have to continue to keep ownership of the domain (renew it when necessary) to be able to benefit from the domain authority and "link juice".