Early stage teleporters used Xen as a slingshot, later ones no longer required it, hence why you had to configure it in Blue Shift but it was never referenced in other games.
The Combine didn't know earth existed until they found it through Xen. So they were then able to tunnel from their universe into ours directly. However they only have long range teleportation, they do not have the ability for localised teleportation. Hence Nova Prospekt to the C17 Citadel, but not to any specific local area.
The rebels have access to localised teleportation technology, hence allowing them to make the short teleport from Kleiners lab to Eli's lab.Mossman posted:
Since the Combine lacked such technology prior to the raid on Eli's lab (Where they would have been able to acquire it) they were dependant on pre-existing transport networks with their own modified vehicles.
The satellite Gordon launches simply enables the Lambdasat network which failed to seal the rift. This network is later reused in Episode 2 by Magnusson to close the combine superportal.