Hey,
Sure, you can set up a company that will own a domain/trademark or other assets and will not receive any income in the beginning.
However, proper setup depends on your goals. If you wish to do so, for privacy reasons you should look for jurisdictions that do not have a UBO register, also you can use a local nominee (director and / or shareholder).
Considering that the company will not have a bank account, the fact that you own it would not be available to anyone.
If you wish to protect yourself from various legal risks related to the business itself (consumer protection, GDPR, etc.) you should also look for specific jurisdictions which would be hard to claim you and which would have lower legal requirements for such kind of business.
You can cover any business-related expenses from your own funds not involving the company. From an accounting perspective, you will have two ways. First, legally company bears these costs (invoices are issued to the company) but they are paid from your account. It would mean that the company becomes indebted to you and one day you can recover the debt from the company (when the company starts making money), this is more favorable from your tax point of view since debt recovery is non-taxable income almost everywhere.
Second, you can take all invoices on your behalf and might not involve the company at all.
Company is such a case you be passive owner, with no activities at all.