Receiving:
You need an MX record for that domain (then it can be used to receive emails). A mail exchanger record (MX record) is a type of resource record in the Domain Name System that specifies a mail server responsible for accepting email messages on behalf of a recipient's domain, and a preference value used to prioritize mail delivery if multiple mail servers are available.
That means if an email is send to contoso.tld the sending server check the contoso.tld domain finds the MX record in the DNS and send the email to the server which is configured behind the MX record.
So if you can change the DNS settings and point the MX record for this domain to your ip address / DNS (rather then the one from the provider where you wish to "order" the domain) then these domains would work yes.
BUT if you have a dynamic IP address (no fixed one) this might cause some issues. See URL below with more infos.
Sending:
For sending you do not really need an domain. You can send also an email with your dynamic IP but most email server will reject that due to SPAM reasons. So please keep also noted that using a dynamic IP for an MX (Exchange Server) isn´t a good idea and no company would do that. For a lab this would be fine for real productive usage I would never do that. For a productive usage you need a fixed IP. You might also wish to check SPF here to avoid issues in your "setup". For a dynamic IP I wouldn´t try to implement that, but you also do not need that in your LAB.
By the way here is a good tutorial for the setup you might wish to build.
P.S.
You might wish to start with Exchange Online. It has similar features but eliminate the need that you need to deal with an self installed server. So for the beginning that might be a good idea to start with eMail services. You can then buy a domain, change the MX record and point that to Exchange Online. This setup would be what some companies use by the way.