Can an elemental be resurrected in 5e

Raise Dead — No. The spell description, at the end of the second paragraph, states:

The spell can't return an undead creature to life. (PHB, p. 270)

Revivify — Probably Not. "[A] creature that has died within the last minute ... returns to life with 1 hit point." (PHB, p. 272) Both Animate Dead and Create Undead have a casting time of 1 minute, so their targets could not become undead before Revivify is ineffective. Similarly, many undead monsters take well over a minute to make (e.g., liches, mummies, vampires). However, there appear to be other ways to create undead in less than a minute, for example, the Finger of Death spell and death tyrant's Negative Energy Cone raise zombies the next turn, a shadow dragon's Shadow Breath creates a shadow the same turn it kills a humanoid, and the wraith has the Create Specter ability. These cases may require DM adjudication. One way could be by Contest (DMG, p. 238; PHB, p. 174) for the soul to take over its undead body due to the opposing magics involved. See generally, Magic Jar spell description (PHB, p. 257). But see, Discussion below.

True Resurrection — Maybe. The spell states, "If the [dead] creature's soul is free and willing, the creature is restored to life ... the spell can even provide a body if the original no longer exists." (PHB, p. 284) This suggests that incorporeal undead are resurrected if their soul is free, as the spell would simply create a new body. For corporeal undead, the question is whether True Resurrection can supplant the undead force with a free soul. (If permitted, this could be done by Contest. See above.)

Ultimately, the DM must decide whether being undead somehow makes a soul not free and thus unable to return to life. See Discussion below.

Discussion

Although not explicit in its description, Revivify probably does not work on an undead creature without a body given that it requires "touch[ing] a creature that has died within the last minute." (PHB, p. 272) According to the Monster Manual, a specter is a "bodiless spirit" (p. 7) and has Incorporeal Movement (p. 279) so True Resurrection may work by creating a new body. However, the specter's description suggests that it is the soul itself—"Some are spawned when ... the touch of a wraith rips the soul from a living body" and "its only end the oblivion that comes with the destruction of its soul" (Ibid.)—implying that the soul is not free to be restored to life.

If a creature is killed by the spell Finger of Death, "it rises at the start of [the caster's] next turn as a zombie." (PHB, p. 241) According to the MM, "Once turned into a zombie, a creature can't be restored to life except by powerful magic, such as a resurrection spell." (MM, p. 315) Reading way too much into this, "resurrection" here is not italicized so it is not the specific Resurrection spell. However, it does say "powerful magic" and a 3rd-level spell like Revivify is not powerful magic, so I don't think it would work; but, True Resurrection is the most powerful resurrection spell and might depending on your DM's interpretation.

Yes.

As you rightly say, those spells contain no such restriction on bringing back a creature who has died of old age.

In the case of Raise Dead, the creature retains their original body and their original age. Therefore they are at risk of dying of old age again in the near future. The exact methods of determining how and when a creature dies of old age are up to the DM.

In the case of Reincarnate, the creature has a fresh body created for them which is "adult". They are not in danger of dying from old age again in the near future.