I notice in your two first examples that you attempt to translate “to become” into «venirse» which, right now, I cannot think on an example in which this is correct. English “to become” is better translated as «volverse», «llegar a ser», simply by «ser» when context is enough to show a process, or by a verbing of the adjective.
So, in your two examples, it is better to say in Spanish
Quiero ser abogado cuando sea grande.
Día a día me desencanto un poco más con el mundo.
- A few notes: note «cuando sea grande» or a more literal «cuando crezca» instead of «cuando sea adulto». Your choice is not wrong, but for me «grande» sounds more idiomatically correct than «adulto».
Additionnally, “day by day” translates just into «día a día».
Your third example, while a little odd, is correct: for example, if your mother is upstairs and you are making trouble downstairs, she would angrily yell:
«¡No me hagan ir abajo para darles un castigo!»
(I'm not sure what «tancancitos» means. Google just point me to another version of this question)
But if she already came down, she would say:
«¡No me hagan venir abajo (de nuevo) para darles un castigo!»
But this is not actually an example of «venirse» as can be noticed that the subject is (tacit) ustedes (second person plural) while «me» (first person singular) is the object. This is not a reflexive construction.
«venir» can be used in most cases where English uses “to come” as non-phrasal verb indicating movement. Note that phrasal verbs such as “to come back” usually have its own verb in Spanish (i.e. «regresar»).
«venirse» can be used as complement to «irse», just as «venir» is complement to «ir» in some cases in which «irse» would be preferred to «ir».
«Voy para la casa.» “I am (already) going home.”
«Me voy para la casa.» “I'm leaving (with the intention to go home).”
«Vine a casa.» “I came home.”
«Me vine para la casa.» “I left (and came home).”
«venirse» (slang) ”to have an orgasm.”