As Unearthed Arcana rules, this writing is a bit unpolished. Here's how I understand it:

You spend your action and gain the psi points; that's all you get from that action (the rule says you gain points that you "can" spend, not that you gain points and spend them). Those psi points are tracked separately from your normal pool.

Whenever you use a discipline later to create an effect, you choose whether to spend your normal psi points or your special psi points on that effect. You can't mix and match for a single use of an effect, but I see no rule that prohibits spending special psi points on one effect and then spending normal psi points on an effect from the same discipline.

If one were to interpret the rules to say that using special points locked a discipline from being available via normal points, there would need to be some rule that explained when that lock expired. Since there isn't such a rule, the rule against mixing point types must only apply to a single use of an effect.

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.