1) Will this affect my SEO?
No it will not affect your SEO. Many domain names are registered and expired in daily basis, so Google already know that you're the new webmaster(From who is data also from your google search console account name, also by checking past content vs new content).
2) How to remove those broken links which do not currently exist. I
marked them as fixed but after few days they again appear.
When Google see 404 link not found status/error code, then Google assume may be that content will come back after some time. So they wait for a long time. The time is not fix, but the search console data is lazzy, Google may already drop that links into search result, but they are lazzy to remove those pages from search console. Click on those broken links and you will see when the last Googlebot crawl that pages.
So when you mark that fix, then Googlebot will come again to crawl those pages, but they again see 404 error page, and hence they report back again to you in search console. To remove your pages instantly from search console use 410 instead of 404. 410 means page is permanently gone and it will never come again, so Google drop those pages in just few weeks.
If you can't set 410 status code, then simply ignore them, Google will drop that automatically after a long time, but don't worry 404 is not harmful for your site.
3) Still, my blog doesn't rank within the first 200 results
Ranking has nothing to do with 404, you need to focus on other important things like making site user friendly, making better content then your competitors, and interact with other like minded people who can share/link your blog post naturally. Also use reddit, because there are so many people, appreciate your content if it is really useful.