What does it mean that Ed exchanged his Gate?
In the context of Ed's final transmutation (in which he exchanges his Gate for Alphonse), the Gate represents his ability to do alchemy, rather than his knowledge of alchemy. In chapter 108, Truth tells Edward:
The Gate of Truth exists within everyone. That is how everyone can use alchemy.
Notice that the emphasis is probably on the potential to do alchemy. (Not everyone has the talent or knowledge to do alchemy in the universe of Fullmetal Alchemist!) Although what is beyond the Gate is alchemical knowledge, per the discussion surrounding Ed and Al's human transmutation attempt, that is not what is involved here. (One way to think of it may be to regard the Gate as necessary in alchemy because alchemical knowledge must be "let out" when there is a transmutation, but I digress.)
Thus, even after his final transmutation, Ed is able to learn about alchemy; he simply cannot put it into practice. (In fact, at the end of the manga, the brothers travel to learn more about alchemy.)
Could Alphonse have exchanged his Gate instead?
In general, one cannot trade one's Gate for something else. (See the discussion at Why can't Roy Mustang bring his eyes back using his gate? for details.) However, it seems possible in theory that Al might have been able to exchange his Gate for something after the connection between his and Ed's was discovered. Some possibilities as to why Al did not carry out such a transmutation:
Transmuting one's Gate for what was lost in one's human transmutation attempt is not necessarily an obvious line of thought. The idea may not have occurred to Ed or Al before Promised Day. Note that Ed only comes up with the idea after Al has sacrificed himself, not earlier, not even when Alphonse was starting to have problems with his body of armor.
Alphonse wanted to recover Ed's limbs; sacrificing his ability to do alchemy for his body would have prevented him from achieving that goal. Alternatively, had Alphonse exchanged his alchemical ability (i.e. his Gate) for Ed's limbs, it might have become impossible for his body to be recovered. Al's body sits at his own Gate, so if the Gate were removed, it might have become inaccessible.