Transitive verbs such as meet in this context do not need a preposition before their object. So, 'I meet him for lunch', is sufficient.
In fact, including the unnecessary preposition is quite common in everyday communication. You will also hear: 'I meet up with him for lunch'. Note, however, that such usage is frowned on in style guides such as Garner's Modern American Usage (628):
Don't use a phrasal verb if the adverbial particle (preposition) is
simply baggage that doesn't add to meaning. Thus, don't say meet up
with if meet suffices. Don't say connect up or divide up if
connect and divide suffice.
But there is one context in which 'I meet with him' is preferred. For example, if you have a predetermined meeting once a week, then using the preposition is the right choice:
As to your second question, argue is an intransitive verb in this context. Intransitive verbs cannot be followed directly by an object. So, the following is ungrammatical:
*The two argued each other for several minutes.
You need the preposition with.