Yes, you can register such a domain name. But make sure you follow some tips while choosing a domain name same as your business name. Like, suppose you have registered a business name as Fresh Cakes Ltd. then you wouldn't want to pick a domain name as nicksfreshcakes.com.

But you can consider other options like :

Hyphens and Numbers can jumble your customers and hence, you need to avoid its use as far as possible. Like, if your juice bar name is 'Juice For All' then avoid using 'juice4all' domain name. It may confuse your audience making them think that you serve 4 types of juice to everyone.
Adhere to A to Z for your effective domain name.

Yes, it does affect but not at a large level. Having a targeted keyword in your domain name is a good sign of having an effective domain name.

It is always better to have the shorter domain name. Would you love to see freshcakesbestinlondon.com as your domain name? It's confusing and difficult to remember right?
Same your visitors will feel about it, so keep it short. According to studies, it is found that the ideal length of a domain name is eight characters, but not every brand will be able to achieve it. So, at least make sure that you choose the name that satisfies this condition.

No, there are other options also available. E.g. for the non-profit organisation (organization) you may register a .org domain name. You can also choose the domain names as per the country code like, for India .in is available likewise, for the UK you have .co.uk You can search such different helpful extensions for your business.

Yes, domain registration lasts mainly from one to ten years. So, it is required by you to note the expiry date of your domain name and renew it before it expires.