I try to change the URL address from https://company.sharepoint.com - I using SharePoint Online (Office 365). Can Microsoft change this address? Or can I create address/record in my private domain, let's says sharepoint.mydomain.com that will be direct to https://company.sharepoint.com?
I have make a mistake in https://company.sharepoint.com this address and want to change for something different.
Did one way to do something like that is buy a SSL certificate?
This isn't possible with private site collections on SharePoint Online currently.
This is because of the SSL requirement (in the E plans) for private site collections. There isn't a way to upload an SSL cert to SharePoint Online so that it can serve requests for that domain and HTTPS.
You can map your vanity domain to your SharePoint Online public web site however (because it doesn't use HTTPS).
If you are accessing that URL from outside your domain, you will need to buy an SSL certificate. You won't have to do this if you use the http protocol. I would take a look at any of the SSL certificate sites that offer SSL certs to meet your needs on this.
Otherwise if you don't care about SSL you can change that URL to whatever you want with some DNS changes and changing the URL in the AAM settings in Central Administration.
