Residency Card, Business bank account, Driver License,

It's been quite a number of years since I was last in Oman and things have changed considerably .

As already advised you will need to get yourself a PRO to do all the face to face admin but at some point I dare say the Ministry Of Manpower will need to see your credentials and even have to be approached by a potential employer, just to get permission to make an approach to you with regards to employment, never mind getting the go ahead to do so and then when you get the job, there's the medical ( or was) that's a preliminary to getting your residency card cum Green card

There use to be and may well still be the British Businessmen's forum, that met once a month that could advise you more ( try the British Embassy or British Council ( use to be in Medina Qaboos) for details.

You own Embassy ought to be able to advise you, with regards to the above, and as a US citizen you may need to exchange your US  driving permit for an Omani one, with out the need to take a test.

Again, try your Embassy for advice on this.

Also, you make mention of  having your own US/Omani company.

Unless things have changed, you will need an Omani sponsor / partner , so I strongly advise you to check this out.