Yes, you can use any and all psionic abilities while wearing heavy armour. It's difficult to prove, but the article doesn't mention anything about armour constraining psionics. Similarly, the handy Sage Advice site built a collection of Mearls' tweets on psionics, and none of them mention anything, either.

Note that if you wear armour that you aren't proficient in, you will suffer the normal penalties of non-proficiency:

If you wear armor that you lack proficiency with, you have disadvantage on any ability check, saving throw, or attack roll that involves Strength or Dexterity, and you can’t cast spells.

It's worth noting (on top of that first note), that this won't really affect your psionics at all, since the attack rolls you make with psionics are Intelligence-based, not Strength- or Dexterity- based.

The Psionic Point limit is a hard cap for a single expenditure

No, you "can't do this all at once" since nothing costs more than 7 points. Since you need to still adhere to the "as an action" or "as a bonus action" limit, the standard action economy applies.¹

None of the specific wording in the Psionic Mastery description offers an exception to the hard cap on how many points you can expend to activate a psionic discipline (one instance of a psionic discipline). You will note that none of them costs more than 7 points, so you'll typically be able to do two things, at least, from that bonus point lump this ability offers.

You'd need to use some of the points on one action, and the rest either on another action, or a bonus action. Once you've activated this, those points are yours to spend until you finish a long rest.

As written on page 4 of the UA document (Psi Limit):

There is a limit on the number of psi points you can spend to activate a psionic discipline. The limit is based on your mystic level, as shown in the Psi Limit column of the Mystic table. For example, as a 3rd-level mystic, you can spend no more than 3 psi points on a discipline each time you use it, no matter how many psi points you have.

That's not a problem, since the Psionic Mastery points can be used for more than one thing, and you can stack various psionic abilities that are eligible which is not something you can do with your regular Psionic Point budget.

Example 1:

You use an action to establish this bonus pool of points, 9, and you then use (and stack) two different instances of a discipline (Telepathic Contact (p. 26)) Occluded Mind (2 points) and Psychic Domination (7 points). That uses up all of your points, and each of them cost 7 points or less. Your psi limit topped out at 9th level, at 7, and you didn't not earn Psionic Master (1/day) until 11th level.

You can use all 9 points on one discipline, or you can spread them across multiple disciplines. You can’t also spend your normal psi points on these disciplines; you can spend only the special points gained from this feature. (p. 9)

Example 2:

You spend 2 points on Occluded Mind (as above) on one opponent, and the other 7 points on a different area near you on Psychic Crush (P. 24) to damage a small crowd of enemies.

Example 3:

You spend 2 points on Occluded Mind (as above) and as a bonus action spend 7 points on a 5th level spell slot (Arcane Dabbler, p. 8) that you will retain "until you use it or finish a long rest."

¹ Ruling territory here, since on page 3. it states that "psionics is a special form of magic distinct from spell casting" but it doesn't say that it has a different action economy. The only 'special' feature is the ability to stack concentration effects when using this particular feature of the class).