Yes, do that! And note that on a number of email service providers you can add both domains to the same mailbox.

Just don’t use a domain name containing your surname - register a different one.

how to deal with whois lookup privacy?

Any decent domain registrar provides WHOIS privacy for free (just note that that might not be available for .eu domain names). Additionally you can always use fake data on your account, but in this case you might lose your domain if they decide to check it.

other issues I may run into if I chose my own domain

When registering a domain, always look at the renewal price: often the first year costs relatively low, but after that there’s a steep price increase.

As for an email hosting service, in addition to the ones you specified in your post, I suggest you to look at migadu.com.