Right, it's a new set of books with no history, and a new EIN. 

The value of the assets being purchased should be raised on the start date. It's probably the tax value in the previous company.  The offset is equity, or maybe the new owner will contribute cash into the new bank account, to pay for the assets.

Goodwill asset, for the difference between total asset value, and purchase price.

What are the assets being purchased?  Depreciable assets I see, so the tax book value would be easiest, unless you want to have different book and tax values.

You need a CPA to discuss this with.

It's good if the start date is 1/1 as that will make payroll set up much easier, as there will be no YTD to enter

"Should we create the new company and import the lists/COA using QBO, or instead using QB Desktop then exporting to QBO?"

First export from desktop to Excel. Then import into QBO from Excel, after ensuring the data layout is correct, and deleting the dormant records.  There is a sample file in the QBO import screen, to show you how the data should look.

Another option is to create a copy of the file in desktop, without any transactional data, and use the conversion tool to create a new QBO file. So you will have the lists of customers,vendors, items, and accounts.  There is no batch delete in QBO, so you wont be able to clean the lists, unless you can do that in desktop.

"Will we be able to export custom invoice templates from QB Desktop, complete with logo and formatting?"

No