'Next Monday' or 'on next Monday'?

If you're asking the specific question regarding the preposition 'on', as to whether 'on next Monday' is valid, then I would say no it isn't, you would never say 'on next Monday'. You would always say 'I will send you the proposal next Monday'. Alternatively you could say 'I will send you the proposal on Monday'.

Otherwise the question is quite open-ended as usage would vary depending on the preposition in question. Taking the preposition 'for', you could say something like 'Will you be ready next week?' and it would be valid to add the preposition as in 'Are you ready for next week'.

You can do something on Monday or next Monday or every Monday.

I found a good reference here

https://www.englishclub.com/grammar/prepositions-at-in-on-time.htm

You don't use 'on' when there is a 'next'.