Travel from NRT to Osaka after landing in NRT

We are flying from the US reaching Narita around 3 PM and planning to travel to Osaka for transfer to Koyasan by train the next day. After reading this forum I have gathered the following:
1. We can fly (to KIX or ITM) or travel by train (Shinkansen). Which option would be the best?
2. Since we are already at NRT, will flying be more convenient? If so, do we fly to ITM or KIX if the plan is to take train next day from Nankai Namba Station for Koyasan.
3. Flights to KIX are much cheaper than ITM, but will the budget airlines flying to KIX allow our bags from international flight?
4. How much time do we allow at NRT for entry, baggage claim and to transfer from international to domestic flight?
5. How much more difficult will it be to take Shinkansen to Osaka instead of flying after such a long flight from the US west coast (since Shinkansen my be an unique experience). But it may cost more than flying to KIX.
6. What would be the best optimum location to stay overnight in Osaka?
7. Another option would be to stay in Tokyo and travel the next day. If so, what would be the optimum location to stay in Tokyo.
8. Our goal is to reach Koyasan the next day before 3:00 PM.

Lots of trade-offs without any clear answer. Please advise.

1. We can fly (to KIX or ITM) or travel by train (Shinkansen). Which option would be the best?

Flying into KIX would be best. It is likely to be cheapest and fastest. It is also a relatively good starting point for going to Koyasan. Don't fly into ITM. It is not convenient for going to Koyasan.

3. Flights to KIX are much cheaper than ITM, but will the budget airlines flying to KIX allow our bags from international flight?

They will allow them, but if you have large bags you will likely have to pay additionally. If you buy the domestic flight in combination with your international flight, then you may be able to get better bag options (Jetstar flights are code-shared with JAL, and some are also code-shared with American and Qantas).

4. How much time do we allow at NRT for entry, baggage claim and to transfer from international to domestic flight?

You will also have to go through immigration and customs. I recommend 2 hours.

5. How much more difficult will it be to take Shinkansen to Osaka instead of flying after such a long flight from the US west coast (since Shinkansen my be an unique experience). But it may cost more than flying to KIX.

It is not really difficult if you do a little research about train tickets in advance and know what to expect. It could be the better option in case you don't have a convenient flight connection (time-wise), because you have train connections every 30 minutes.

6. What would be the best optimum location to stay overnight in Osaka?

I recommend KIX or Namba, from where access to Koyasan is easy.

7. Another option would be to stay in Tokyo and travel the next day. If so, what would be the optimum location to stay in Tokyo.

Near Tokyo or Shinagawa Station.