Is there any guide or advice on how to upload NFT marketplace boilerplate to hosting and connect a domain? Do I need to change some files of the React application or just build and upload the folder?
You can not connect yet your Moralis server domain to a custom domain. You can upload the build folder to Moralis static hosting to see if it works ok.
We are working on supporting custom domains.
How can I use the moralis-based NFT marketplace without a domain?
My client has a domain and he wants his site to open on it.
And if i can’t use a domain - how to host the site somewhere other than a local server? 
you can host the site wherever you want, on your custom domain. it should work the same as it works on local server