You can smite divinely while enraged.

There's no spell named divine smite; you're not casting a spell when you use that class feature. It's a feature triggered when you hit with a weapon attack.

This, by the way, is another way you know it's not casting a spell: you're (most-likely) taking the Attack action to make your melee weapon attack, rather than the Cast A Spell action. It's rare--and explicitly spelled-out circumstances--where you'd make an attack but end up casting a spell.

Yes, a mystic's psionics would be suppressed by an antimagic field

Antimagic field says:

Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it.

So anything that is a spell or other type of magical effect will be suppressed.

The Sage Advice Compendium provides a handy checklist of questions to ask to determine if something is considered to be magical for the purposes of antimagic field and related effects. One of them says:

• Does its description say it's magical?

If this answer to this question is yes, then that thing is considered magical and would be affected by the antimagic field.

Psionics is described as being magical

So, looking in version 3 of the Psionics UA we can find passages such as:

Psionics is a special form of magic use, distinct from spellcasting.

and

Psionic talents and disciplines are the heart of a mystic’s craft. They are the mental exercises and psionic formulae used to forge will into tangible, magical effects.

and

Psionic disciplines are magical and function similarly to spells.

Thus, it is pretty clear that the use of psionics is indeed magical and would be affected by an antimagic field.

Note: spells and magic are commonly equated when, in fact, there are many magical abilities and effects that have absolutely nothing to do with spells. So, to be clear, psionic abilities are not spells even though they are magical and even though some of them act a lot like spells. Things like dispel magic, for example, will not work on psionic abilities as a result of this.