Does it make sense to train multiple classifiers for multiple partitions of data?

It can make sense, but more commonly I think it doesn't.

When you split into two models, those two models can't help each other: any common trends between your sub-populations will have to be modeled by both models, and so probably won't be estimated as well as by one larger model. But, if those populations are really different, that can be a benefit rather than a detriment, especially for low- capacity models. As for tree models, note that splitting into two populations would be exactly equivalent to splitting on that variable as the first split in every tree (extremely unlikely, especially if you're sampling columns).

Yes, there is no need for you to manually split the dataset into two parts and train separate classifiers for them.

That's something that decision trees (eg. Random Forest), will do, as you say, under the hood.

The manipulation that you could do on your dataset is probably the dimensionality reduction - but, this is also done, informally, by decision trees. See "Random Forests / Ensemble Trees" section here.