Hi Chowsi,
Welcome to flyertalk!
First off, late January and early February are putting you there during the rainy season. Many people aren't bothered by that, but if you are, you might want to wait and go later in the spring.
Also, you probably already know this, but it is worth mentioning. Borobudur has strong Buddhist roots but the island of Java itself is strongly Muslim. Morning calls to prayer, etc. It's very special.
1) I would recommend 4 nights at Jiwo if you can swing it because you are taking a separate flight to get there. Kila and Dari are in comfortable driving distance from one another so your stays can be shorter.
2) Their packages are always an excellent value so Bali and Beyond would work well for you. Aman manages all transportation and transfers once you are in Indonesia (car transfers are free and they will charge you for planes as necessary). They are especially good at getting you from one place to another in style and comfort.
3) Nothing wrong with taking your mother as long as you don't mind sleeping in the same room. The regular rooms at Amans are large but they typically don't have dividers. Aman will arrange separate beds.
4) Food costs are pretty reasonable considering these are luxury hotels. And they can send you the activity sheets ahead of time that have prices on them for budget purposes. Generally speaking, Aman does not nickel and dime. You pay a high price for the room but that's it.
5) Have each property email you their list of available activities. All phenomenal!
6) I (and many others) have detailed trip reports on this forum for each of the properties you are considering. Check them out and feel free to PM each of us with individual questions. No, Jiwo does not offer the elephant trip to Borobudur anymore but they might still do the elephant painting.
7) Yes, you can PM DavidO.
What an exciting trip! You have a very lucky mother! Keep us up to date on your plans.