How to choose the name
Short, pronounceable over the phone and without hyphens. If you have to dictate "with a hyphen in the middle", the name is not good. Avoid numbers and deliberate misspellings. Check how it looks joined up, because some names read differently once the spaces are gone. And check the name is not taken as a trademark, because losing a domain to a trademark claim after two years of work is an expensive lesson.
Which extension
If you work in one country only, the country extension is legitimate and signals a local business. If you sell abroad too, .com is safer because it does not tie you geographically. Do not migrate from .com to a country domain for SEO reasons: a ccTLD narrows geographic targeting rather than helping, and a domain migration always costs some visibility. Other extensions are fine when they make sense for the brand, just know that some buyers still read .com as more serious.
Prices
A domain is paid annually. A .com is usually ten to fifteen EUR, country domains often around twenty. The first year is often promotional, so always check the renewal price. If somebody asks materially more, you are paying for intermediation.
The rule that prevents the worst outcome
The domain must be registered to you or your company, with your email as the owner contact. Not to the agency, not to the person who built the site. This is the most common way businesses lose control of their own address, and it only surfaces when you want to move to another supplier. Turn on privacy protection and automatic renewal too, because an expired domain can be bought the same day.
With us the domain and accounts always belong to the client, which is also covered in the piece on choosing an agency.