Stalnaker's conditional is defined by a set of axioms given in his paper "A Theory of Conditionals". The idea behind it is to express under what conditions the consequent would be true, were the antecedent to be true. It is intended to cover both indicative and counterfactual conditionals. It is usually described as a variably strict conditional, since its truth depends on what holds in relevantly close possible worlds, but not in all possible worlds.
Roughly speaking a conditional 'if A then B' holds when, in the closest possible world in which A holds, B also holds in that world. Or we may state it as that the closest possible world in which A and B hold together is closer to the actual world than the closest possible world in which A and not-B hold. As you correctly say, some familiar forms of implication that are valid for the material conditional are not valid in general for the Stalnaker conditional.
You mention hypothetical syllogism, i.e. that A > B, B > C, entails A > C. Hypothetical syllogism can fail because it may be that in the closest A-world, B holds, and in the closest B-world C holds, but in the closest A world C does not hold. Examples of this occur when C is typically or nearly always the case when B is true, but A preempts C. Stalnaker gives as an example: "If Hoover were a communist he would be a traitor; if Hoover were born in Russia he would be a communist; therefore, if Hoover were born in Russia he would be a traitor". Ernest Adams gives an example like this: "If Brown's party loses the election, Brown will resign as President; if Brown dies his party will lose the election; therefore, if Brown dies he will resign as President". Here is another: "If Mary spends lots of money on luxury goods she will be poor; if Mary wins the lottery she will spend lots of money on luxury goods; therefore, if Mary wins the lottery she will be poor".
Note in each case the A term preempts acceptance of C. Another way to generate counterexamples to hypothetical syllogism is to use uncertain propositions. A high value for P(B | A) together with a high value for P(C | B) does not entail a high value for P(C | A). Hence, it is possible for it to be highly probable that 'if A then B' and highly probable that 'if B then C' but highly improbable that 'if A then C'. Look for cases where it is highly probable that C given B, but highly improbable that C given A and B. Again, the A acts to preempt C.
You also mention that unlike the material conditional, A does not entail ¬A > B. This is so because for the Stalnaker conditional A being true at the actual world does not guarantee that in the closest possible world in which A is false, B is true. Counterexamples to this form of inference are two a penny.
Another common form of implication that fails is or-to-if, i.e. that 'A or B' entails ¬A > B. Dorothy Edgington offers an example like this: Let A be "I'm not shot in the next five minutes", and B "I'm not injured in the next five minutes". I consider 'A or B' to be true, or at least highly probable. But ¬A > B would be "if I am shot then I am not injured" which is false, or at least highly implausible. Or another: "It will rain or it will snow in London in July this year". Highly likely since it is rare to go a whole month without rain in London, even in July. But not, "if it doesn't rain in London in July it will snow".
Robert Stalnaker, "A Theory of Conditionals", 1968, in Studies in Logical Theory, ed. Nicholas Rescher, also reprinted in Stalnaker, "Knowledge and Conditionals, 2019, Oxford.
Ernest Adams, "The Logic of Conditionals", 1975, Reidel.
Dorothy Edgington, "On Condiitonals" 1995, Mind, Vol. 104, pp 235-329.
Dorothy Edgington, "Validity, Uncertainty and Vagueness", 1992, Analysis, Vol. 52, pp. 193-204.
