Since Alkahestry is used via the dragon's pulse while alchemy used in Amestris was powered by the philosopher's stone, could Edward learn and use Alkahestry after he had given up his gate of truth and lost his ability to use Alchemy?
In all likelihood, he can learn it but not use it.
I say this because Alchemy and Alkahestry mainly differ in their sources of powers. As mentioned on the Wikia article on Alkahestry:
Alkahestry refers to the slightly different form of Alchemy used in the country of Xing. Alkahestry differs from Alchemy in both its practice and its goal. Whereas Amestrian Alchemy claims to have its roots in the energy of tectonic shifts and practices manipulation of matter toward scientifically practical ends, Alkahestry is centered on a concept called the "Dragon's Pulse" which speaks of the Earth itself having a constant flow of chi (life energy) which flows metaphorically from the tops of mountains down to the land, nourishing everything it passes with that energy as does blood coursing through the veins.
Although never explicitly stated, it is very likely that both Alkahestry and Alchemy make use of the Gate, and their difference lies only in their source of power and their practices. The above excerpt even mentions it as a "different form of alchemy".
Alkahestry uses energy flows of the planet to open the gate. The energy flows and has many different exit points and thus they can use ranged alchemy. The concept of Chi is that the human body has an energy flow and that it also has many different exit points.
Using this concept, Alkahestrists are capable of a higher level of medical transmutation than Amestrian alchemists - transmitting chi through the pathways of the human body to cure mild ailments as well as minor injuries - and can even project their transmutations over long distances and wide areas by accessing and guiding that current to their own means with Purification Circles and Alkahestric markers, a function of which Amestrian Alchemy is altogether incapable.
Amestrian Alchemy uses tectonic rifts as their source of power.
But really Father's Philosophers Stone is limiting that source of power. Its the reason why Father could amplify the Freezer's Alchemy (Episode 1), turn off Alchemy (but not Alkahestry) and why Scar's transmutation (after Bradley died) gave everyone increased power (it stopped the inhibitor of Father's Philosophers Stone).
Because of the country's history of war, their alchemy was developed with combat use in mind. Some medical alchemy was developed (proof being Marco and others) however it was never specialized and Alkahestry's medical alchemy was "better".
In conclusion: Since both of them make use of the same concept/power, the different lying in their sources and their specializations, it's possible for him to learn Alkahestry but not use it. It's similar to what happened with Alchemy. Ed can't use it at the end, but there's nothing stopping him from learning it. He even departs to the West to learn more about alchemy. As someone on Reddit put it:
He can still physically do the motions for both, it's his body that can't make the final connection. To draw an analogy, it's like someone who is paralyzed: Their legs are still there and the nerves are still there, but they aren't connected anymore.
If I've understood alchemy and principles behind it correctly, then - no.
Alchemy in any of its forms is science of changing world around you. The question is - how you change it. If you remember island episode, answer is given there. Everything in world is connected. You change one thing - it changes everything connected to it in chain reaction. Obviously, human is part of this mechanism. Alchemy transmutation is basically some sort of change inside of human itself, which then leads to changes in real world. Circles, tattoos etc are just a method to concentrate on change you want to do.
Now, about gate of truth. As I see it, it is actually representation of connection between human and world. Once human undergoes change in result of transmutation, gate translates it to the world.
Edward destroyed this gate, effectively removing his own connection to the world (in terms of alchemy). So, I think, it is not possible for him to do any type of transmutation.