Eladrin do, they have a racial ability called Fey Step which lets them use a bonus action to teleport up to 30ft (With the added options I mentioned earlier). The only other elf race that gets a natural teleport ability is the Shadar-Kai (Elves tied to the Shadowfell and the Raven Queen), who can teleport as a bonus action and at higher levels gain resistance to all damage for a short time after.
Other Elven subraces don’t inherently have the ability to teleport but do have other abilities, usually some combination of Spellcasting, Weapon Proficiency or Supplemental Skills. The only other thing close to a racial teleport is High Elves can take a racial feat to get the Misty Step spell (Teleport 30ft) with one free cast per short/long rest
Pretty much, except for the racial feat I mentioned since it’s a feat.
In 5e, you tend to get your Racial ability stuff at character creation, save for some more powerful stuff that unlocks down the line. For instance, the Teleport abilities I mentioned for the Eladrin and Shadar-Kai are things they have out the gate, but the added effects of those teleports don’t come until a few levels later.
It’s the same for a few other races where they will have some starter thing that is improved later on (IE, Tieflings start with a set Cantrip and level 1 spell at the start, and at levels 3 and 5 get higher level racial spells, which are all determined by the circle of hell your bloodline is tied to), but most stuff is out the door from character creation