Not really. It's pretty explicit what you can and cannot do.
"You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. However, you can't use any of your special senses, such as darkvision, unless your new form also has that sense."
Is your new form capable of limited shape-shifting? If not, then you can't do so in that form.
That depends.
For example, can a changeling's ability to change a voice still work? That's limited shapechanging.
Likewise, the ability to change hair color abs other options from that race would (I think) work.
Sprouting limbs is, I think, different, but I think that's also debatable.
At higher levels, I would say that elemental forms would allow the sprouting of limbs.
For animal forms, my first instinct would be to say that you can't just pip out an extra set of limbs. Thinking about it more though, I'm not 100% sure. A dragonborn's breath weapon continues to work, so that suggests that some innate abilities which aren't a function of the new form -even when the animal cannot normally do those things- still work.
How much does D&D shape-shifting and wildshape change the internal or biological capabilities of the user?
The reason I'm asking is because it might be possible for a form that doesn't normally have hands to have hands -which matters for some of the game rules.
Edit: Further thoughts...
If it were a flight speed or some other ability which made use of an aspect of a physical form which no longer exists (like wings), I think it's obvious that the new form cannot use something which ceases to exist.
However, in the case of the Plasmoid, the limbs are not something which exist and then do not. Instead, they are something which do not exist by default and are created by some manner of innate ability.
I think that innate ability relies on the ooze nature of the plasmoid form, but I could also see an argument for it being something internal and innate like a breath weapon.