In Doctor Who (the old series) why can't they get back into e-space? Does it still exist? Did it have any relevance to the alternate universe in the newer series?
They can't get back because they only got there by accident and the 4th Doctor isn't able to pilot the TARDIS very well, but it does still exist: he does go back eventually. It's not related to the parallel universes / timeframes that the 9th through 11th Doctors experience.
The TARDIS originally entered E-Space through a Charged Vacuum Emboitment (CVE) created by the Logopolitians. After the events in Logopolis, only one CVE was left (and the Logopolitians were no longer around to make more), and this did eventually close (in Timewyrm: Apocalypse). (In The Invasion of E-Space we do meet another race that can create CVEs).
Travel through the CVE wasn't a stable process, and the 5th Doctor says (in Earthshock) that it would be extremely difficult for the TARDIS to travel to "negative coordinate space", and that this wouldn't occur accidentally. This effectively means that E-Space was cut off from N-Space at least for the Doctor at this point in this timeline.
However, the 7th Doctor, better at piloting the TARDIS, was able to travel between E-space and N-space, and brought Romana back (in Blood Harvest).