Hi Tymen,
Great questions, to answer them:
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Yes, you should try to change your backlinks to match your HTTPS adress but don't sweat it if you can't change all backlinks.
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Try to avoid 301 redirecting internal links. There are many great guides to do HTTPS migrations in Wordpress and doing complete rewrites from HTTP to HTTPS.
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It might affect your rankings short-term, but that's not necessarily the case. We've enables HTTPS on multiple websites without any noticeable drop. Be sure to check out these great guides to help you out:
https://moz.com/ugc/increase-conversions-by-fixing-https-errors
https://moz.com/blog/enabling-https-without-sacrificing-web-performance
https://moz.com/blog/seo-tips-https-ssl
Hope this helps!