So the questions:
1. Will the new phone come with a sim installed?
Yes, the Verizon Moto X's come with Verizon SIM cards, including the Verizon Moto X DE
Moto X Developer Edition for Verizon - A Google Company
2. Can I just call verizon business support to activate the new phone on my existing line without involving the company admins?
The SIM card that comes with the phone has to be associated with your phone number through Verizon either online (which requires that you have the account User ID and password) or by phone (which requires you to be listed as an authorized person on the account and to have the account passcode). When you call Verizon, they will ask you your name and for the account passcode and if you are not authorized on the account or don't have the account passcode, then no, you won't be able to do it.
3. Is it easier or better to do this at a corporate verizon store?
You won't be allowed to do it at a store either unless you are authorized on the account to do it. They will ask for a driver's license from you in store. Over the phone they will ask your name and for the passcode.
4. Is there any way to use both phones on the line without calling Verizon to activate and deactivate each time?
There are SIM card adapters and SIM card cutters that you can use to make the SIM card you have (which is already associated with your phone number) fit into both devices. The GNote 2 has a micro SIM card and the Moto X has a nano SIM card. You could buy a nano SIM cutter to cut the micro SIM down and then a micro to nano adapter for use in the GNote 2.
However, when you switch between phones by switching SIM cards, it will show up in the account that a new phone is on the line. Everytime I switch SIMs between phones, after I log onto Verizon, I can see that I did that because it shows me what phone is currently on the line. Also, if you have asurion insurance, it only applies to the phone that is active and if you keep switching back and forth, I am not sure how that works with the insurance coverage.
And the account admin can set up the account to have email or text alerts to be notified of this sort of change. So if your company doesn't allow you to do this, you could be found out quite easily.
5. Is there a way to do this swapping online without being an account admin? That would be best case.
Yes, if the account admin has authorized you to do it and given you the online User ID and password. If you obtain those without the account owner's permission, I guarantee you that you will have broken a law and could be potentially subject to criminal prosecution no matter what state you live in should the account owner choose to pursue criminal charges. There could be civil fines, too.
Remember - you do not own that line; your employer does. Unless you are authorized by your employer to make changes to the account, including changing the phone on a line, I urge you not to do it or you could find yourself in trouble with your employer and depending on what you have done, the law, both criminally and civilly.
6. Actually, best case would be to just swap SIM cards but given the different sizes I doubt that is possible. Any work arounds?
Yes, see above with the SIM cutter and SIM adapter. However, please also keep in mind everything else I said - you do not own the line, and swapping phones like that not only will show up online but there could be alerts set up on the account, too, that will notify admin of the change. And especially keep in mind that if you are not allowed to do this, you could lose your job.
It's just not worth it. Ask for permission to do this and if they say no, then don't do it. Or pay for your own verizon account to use the Moto X on.
To reiterate - that line is not yours. It is the property of your employer so treat it like you would treat any property of your employer's and don't do something with it you are not authorized to do.