A multiple choice on formal languages, Turing machines and automata

Question

$ \newcommand{\lang}{\mathcal L} $ I have been given the following claims about formal languages, that are either true or false:

1. If a language $\lang$ can be accepted with a finite automaton, it can also be accepted with a deterministic Turing machine.

2. If a language $\lang$ can be accepted with a pushdown automaton, it can also be accepted with a deterministic Turing machine.

3. If a language $\lang$ can be accepted with a non-deterministic pushdown automaton, it can also be accepted with a deterministic pushdown automaton

4. If a language $\lang$ can be accepted with a deterministic Turing machine, it can be accepted with a non-deterministic Turing machine.

My thoughts

1. This is true, as all finite automata $A$ can be made deterministic, after which it is easy to construct a deterministic Turing machine $T$ to simulate one: simply use $A$ as the control unit of $T$, maybe adding transitions to the appropriate halting states of $T$ via the empty symbol $\varepsilon$ (not to be confused with the empty string $\epsilon$), once the accepting states of $A$ are reached.

2. This is probably untrue, as not all pushdown automata can be made deterministic, and we would not be able to use a non-deterministic automaton as a blueprint for a deterministic Turing machine. A non-deterministic Turing machine might do here, but not a deterministic one.

3. This is definitely untrue, as deterministic context free languages are a proper subset of context free languages.

4. This seems to be true. I am aware the implication works in the other direction: if a language can be accepted with a non-deterministic Turing machine, there is a deterministic machine that recognizes it as well. The idea was that you could give a description of a non-deterministic machine to a deterministic one as a string, which would allow recognizing the non-deterministic language deterministically. Intuitively the other direction being asked here should hold trivially. Simply add empty transitions that maintain the configuration of the machine to the deterministic machine.

My question therefore is, am I right in my answers?