No

That would be alarming, wouldn't it? I got the same impression from the Healer description. But digging around a bit, I found this text on p. 84:

If your connection lists a single variable-level spell... you gain that variable-level spell when you first get your connection spell for each level. The next time you gain a connection spell, add the next higher level of the variable-level spell to your list of spells known. You immediately lose the lower-level version of that variable-level spell and replace it with the listed connection spell for that level.

Huh?

It's still not super well-explained IMO. Let's look at an example. When you hit 4th level, you "first get your connection spell for [that] level", so you get mystic cure as a second-level connection (think domain) spell. You then "immediately lose" the first-level version you've been using and replace it with lesser remove condition. In other words, the connection spells of a Mystic 4 are lesser remove condition and mystic cure 2; a Mystic 7 gets lesser remove condition, remove condition, and mystic cure 3; and so on. The result makes sense; it's just awkwardly phrased.

(Note that at higher levels, you can still cast mystic cure at a lower level if you want to. So it's good that you get an alternative as well.)

Armor worn under armor has no rules for its functioning, so Second Skin wouldn't work for extra slots, barring houseruling. However, since the power armor question was closed as a duplicate, I'm answering it here instead.

Unlike Second Skin, there are specific rules for how power armor and light armor interact.

If you’re wearing light armor while in powered armor, you gain the higher of the EAC bonuses and the higher of the KAC bonuses between the two suits of armor, and you take the worse maximum Dexterity bonus and armor check penalty.

Since both have an effect, you should be able to utilize any non-stacking upgrade slots in both, with some restrictions. Since you use the power armor's move, and not your own, things like jump jets would not work in the light armor.

Specifically regarding force fields, things get a bit fuzzy. Pathfinder had rules about Temporary HP not stacking, and only the higher value applying, but I have been unable to find similar rules in Starfinder. So unless there's something specific I'm missing, shield upgrades fall into the category of "untyped bonuses" for stacking purpose, and would both function fine, with the outer shield losing temp HP before applying to the inner one.