Learn how to be agile first.
I think there is a problem with adopting agile for the first time along with an enterprise system? If you had a startup would you use something like QuickBooks or some large Enterprise accounting system like SAP? Seems like you should figure out how to handle the accounting and other systems for your company first with something more appropriate. I would recommend starting with a small project going agile that has enough importance so everyone takes the process seriously, but not so super mission critical and timely which will probably suffer by trying anything new.
Question 1: When should one choose an enterprise agile software
development framework, and when should one just let their agile
Start with a small group. In my experience, agile like anything else should start with something established when possible. Why reinvent the wheel especially if you don't know how to invent a wheel? A lot of agile methodologies have much in common. There will be some aspects where many of your team members will look at certain parts as absolute deal killers (e.g. pair-programming). You'll probably be able to weed many of them out. Select a few for review depending on how much resources you have to this process. Sure it would be great if you could pick 10, but may only have enough time for 3.
Yes, there will be constraints, but you can iterate this process (Which is in the spirit of agile.) and make adjustments based on some data and experience and not just pure personal preference. There is the risk of someone being a zealot to a particular process and wants to adhere to it at the detriment of the project, but that can happen with anything. The key here is to not have too many moving parts with too many changes much like good experimental design. If you exclude some aspects too soon, you risk losing the effect of certain parts working together (Pair-programming gets a bad rap, but so does replacing the lone developer on a project.). However, if there are aspects of a particular methodology that is going to be a massive undertaking for your team (e.g. No one has a clue about unit testing.), you may need to give it some time before trying to do everything at once.
Question 2: If choosing an enterprise agile software development
framework, how does one select the appropriate framework to use for
This should be a lot easier once you have adopted and possibly adjusted for your needs at a small scale. It will be easier to put a list of needs. You can ask others about particular types with more specific questions because you will be able to convey your particular needs.