As for putting the landing page on your homepage, please note that landing pages cannot be embedded within pages on your site; rather, they are standalone pages on their own. If you want your ConvertKit landing page to fully take over (as in, fully replace) your blog's homepage, you can set it up on your custom domain — tutorial here39. (Or if you want to still use your domain for it but not have it take over your site, then you'd want to set up a subdomain — more details are included in that tutorial.)
Alternatively, if you just want to embed the opt-in within your existing homepage, I'd suggest setting it up as a form instead, as those are intended to be embedded within pages!
We also have this guide41 on a trick for how to test out your own opt-ins, that you might find useful for testing purposes!
I hope that helps! ☺️