Does a type-1 automaton always have to terminate?
Pushdown automata do not necessarily halt. They are not forced to read input each step, they can also do so-called $\lambda$-instructions where the tape is not advanced.
Then we can have an infinite loop at a certain tape position.

Likewise, linear bounded automata may loop, and do not always halt. However, they can be simulated by a Turing machine, which stores each configuration, and tests for loops. That is not efficient of course, but can be done. In that way we get halting machines, but of a different type.