Is there a way to pass a dynamic from name for SMS sends via API?
Is there a way to track replies to SMS messages but also see the body of the message the reply went to, the body of reply itself, customer id etc. (some kind of a dataview or similar)?
Question 1 - not at this time. Most of the countries do not even support a From Name on SMS.
Question 2 - partially possible:
You can track responses and trigger specific events depending on the context of received message.
Responses would be stored with the number it originated from - so technically you would be able to build an automation with SQL queries that match phone the number with your customer db and SMS data views.
This way you can write output to a new Data Extension with the information you need, like Customer ID, phone number, content received, time received etc