Yes, message execution cannot be persisted across checkpoints (and a newly joining or catching up replica would need that in-progress state to start from). Meaning that at checkpoint heights all DTS executions have to be aborted and then restarted from scratch after the checkpoint.
Yes, the state that the query is executing on needs to be kept around until query execution completes. And although regular (not composite) queries only require the state of the respective canister, the implementation currently holds on to the full (subnet-wide) replicated state (which can easily be tens of GB of extra memory). And then there are composite queries, where you cannot know ahead of time which canisters are going to get called, if any, so you kind of need to hold on to the full replicated state.