Deciding which logical proposition is stronger
The notion of strength of a proposition comes from its power to constrain the model. For instance the proposition $P1 : x = 2$ is stronger than $P2 : x \geq 2$ in Arithmetic. And in propositional notation this can be written as $P1 \Rightarrow P2 $.

The stronger the proposition the harder it usually is to prove. On the other hand, if proven it provides more corollaries.

For instance for a signature with 2 propositional symbols $P$ and $Q$ we have the following partial diagram for the Strength relation.



Now to your examples. $\textrm{False}$ happens to be an extreme case. It can never be satisfied. If you are using propositions to constrain your search space for an answer, then $\textrm{False}$ says: "Call off the search and start again". The other extreme is the proposition $\textrm{True}$. It is trivial. It gives absolutely no information on where to look for the answer. 

Now we shall use this intuition to tackle your questions:

1) True, False

Since $\textrm{True}$ is a trivial constraint, you would not be able to derive anything non-trivial out of it, this certainly means that you would not be able to derive the extreme constraint $\textrm{False}$.


  $\not\vdash \textrm{True} \Rightarrow \textrm{False}$


2) True, True

The propositions have exactly the same strength, they are equivalent


  $\vdash \textrm{True} \Rightarrow \textrm{True}$


3) False, False

The propositions have exactly the same strength, they are equivalent


  $\vdash \textrm{False} \Rightarrow \textrm{False}$


And an extra case

4) False, True


  $\vdash \textrm{False} \Rightarrow \textrm{True}$


Note: there is an issue with saying that proposition $P$ is stronger than $Q$ if $P \Rightarrow Q$ is a tautology since intuitively the relation of being stronger is strict. It would better align with the intuition if we were to say that proposition $P$ is stronger than $Q$ or of equal strength if and only if $P\Rightarrow Q$.