hmm can not really tell you if private or public proxies are better here. I guess using public once will (as always) get them banned sooner, but you have always new proxies here so I don't know, shouldn't matter that much.
well you submit the domain only and you get the data from that. in SER you set it up to get PR (emulated) using domain or subbsomain and it is taking care of it.
With having it checked, it will tread your outgoing IP the same as every proxy. So it will use your real IP as well when performing requests to metric services.
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Please have a look at this article - https://blog.terrykyle.com/which-is-easiest-to-manipulate-majestic-ahrefs-or-moz-spoiler-alert-answers-inside/