Redirect a domain to another host

hello 

The company I work for has a domain which is used with m365 email services, and a website both are provided by hosting company (co1)

the email services and support provided by co1 is fine and no plan to change it yet (fresh contract)

but the website is not good enough

we are creating a better website on a different hosting company (co2) 

I have a  couple of questions:

1-  Is the domain transfer going to transfer all M365 services to the new hosting company? (co2) 

2- As an alternative way does the domain redirecting /Cname or whatever way can cause any problems with M365 email service?? or it is limited only to website usage?

Thanks