how much does it cost?

From $10/y for cheapest VPS and domain. If you have normal provider, who are hot hesitate to give you constant IP and set reverse DNS record, your own computer (starting from Raspberry PI) at home could be used. So, at minimum you have only domain name to pay. Free domain names are not a good idea, since you have a nonzero chance to suddenly lost it without any notification.

Buying VPS and domain names do not require any acknowledgement of info you provide and could be paid with crypto. Of course if you are smart enough to not use all that AWS/Hetzner/Yahoo/wharever corporation crap.

and any guides for it?

Tons of howtos around. Most popular SMTP servers are Postfix, Exim, Courier-MTA. For each you will find numerous howtos and examples on how to setup and run it. All free and opensource.

Also, there are numerous howtos and forums on buying cheap hostings and domains like lowendtalk.com and others where you could find nice offers and discounts.

how much does it cost?

From $10/y for cheapest VPS and domain. If you have normal provider, who are hot hesitate to give you constant IP and set reverse DNS record, your own computer (starting from Raspberry PI) at home could be used. So, at minimum you have only domain name to pay. Free domain names are not a good idea, since you have a nonzero chance to suddenly lost it without any notification.

Buying VPS and domain names do not require any acknowledgement of info you provide and could be paid with crypto. Of course if you are smart enough to not use all that AWS/Hetzner/Yahoo/wharever corporation crap.

and any guides for it?

Tons of howtos around. Most popular SMTP servers are Postfix, Exim, Courier-MTA. For each you will find numerous howtos and examples on how to setup and run it. All free and opensource.

Also, there are numerous howtos and forums on buying cheap hostings and domains like lowendtalk.com and others.