What will I miss if I don’t explore Mother Base

To answer your title - No. I've heard a theory that the reason Big Boss doesn't salute back is because the last person he saluted was The Boss (maybe also why he refuses to salute in the ending cinematic of MGS3)

For the other parts of your question, Punching and tranqing them is all you can do with them - Them thanking you is sincere, it's sparring with them essentially.

And finally, petting DD is something I believe that can only be done out on the field.

When the game notifies you that a fight took place at Mother Base it essentially means that one soldier with the Troublemaker (Violence) trait (that you can view from the Staff Management in the game menu by hovering over the soldier) got into a fight and sent another staff member and ultimately injured them.

To prevent future incidents you have a couple of options the first one is to remove any staff with the trait Troublemaker (Violence) from the organization. Usually this is a good idea for low-ranking soldiers (all skills C or below) but suppose you have a high skilled soldier that happens to have that trait.

Secondly, you can assign another person with the same Troublemaker (Violence) trait in the same skill group. Because there will be two or more Troublemakers with the same trait then they will keep each other on watch, and no fights will actually break out.

Finally, you can also use a staff member with the Diplomat skill. They will help preventing fights from breaking out at Mother Base.

On the other hand this doesn't apply for Troublemaker (Unsanitary); they can be negated since the other Troublemaker with the same trait will negate the effect, but it also depends on your Medical team's effectiveness.

These events cannot be seen take place real-time at Mother Base. My guess is that they are there to encourage you to regularly keep an eye on the staff.