For that you would need something like «Vanno allo zoo dopo che mangio il pranzo.» or «Vanno allo zoo dopo aver pranzato.» ("They go to the zoo after having eaten lunch."). If you want to express that "I" is doing the eating of the lunch, you at least need a «che» between «dopo» and «pranzo».
Because, in this sentence, pranzo is a noun, not a verb. The sentence is not specifying anyone doing the action of eating lunch. If you want "They go to the zoo after they eat lunch," then you would need «Vanno allo zoo dopo che pranzano.» See my first comment on this thread.
No, it should not be. Grammatically speaking, «vanno» is conjugated in the present tense, so "They go to the zoo after lunch." In English, this sentence implies that they always go the zoo after lunch, which the Italian sentence «Vanno allo zoo dopo pranzo.» can also mean. However, you are right that the present tense (especially with the verb «andare») is used sometimes to mean the very near future.