It's a class feature you can do as your action any time you're able to take an action without restrictions on what kind of action (e.g. not a Hasted action).
It's an at-will ability, balanced accordingly. (For example, the previous thing you created disappears if you do it again, so that, rather than a per-rest limit, is what stops you from creating a room full of trinkets.)
So yes, you could fidget with it, maybe even less obtrusively than "doodling" with a cantrip like Minor Illusion which has components S, M (a bit of fleece) which might require you to wave your arms around while wiggling your fingers (as well as brandishing a bit of fleece or your arcane focus), depending on your DM. Minor Conjuration has no listed components, so you could presumably just do it fully mentally, if necessary.
The other requirement is given by targeting:
You also need a hand for it to appear in, or sight to see an unoccupied space on the ground within 10 ft.
If you've been captured and had your hands cut off, and blindfolded, you can't do it. Or you used Shapechange (9th) into a snail or something without hands, and you left the floor so cluttered that there's no unoccupied space...
Shapechange (9th) is on the wizard/druid spell lists. Like druid Wild Shape, it allows use of class features, unlike Polymorph (4th) or Animal Shapes (8th) which replace your game statistics (character sheet) without specifically allowing you to keep class features. So if you want to be a spider on the ceiling of the enemy's hideout, bamfing weird stuff on the floor, like a little statue giving them the finger, embarrassing adult toys, etc., you would need to multiclass into druid, or cast Shapechange, or otherwise be able to use a class feature.
Minor Conjuration isn't a spell
- Using it wouldn't end Invisibility (2nd).
- You can do it while in Gaseous Form (3rd), even though that prevents you from attacking or casting spells.
(But not Wind Walk (6th): The only actions a creature can take in this form are the Dash action or to revert to its normal form.) Also not Meld into Stone (4th): you don't have hands, and can't see.
- It doesn't have verbal or somatic components so you can do it while tied up and gagged, as long as you're not blindfolded or can still conjure into a hand. (But it has to be on the ground, so you can't conjure a series of rocks to fall onto your chains or bonds.)
- A multiclass druid can do it while wildshaped (as discussed above) even without the druid 18 beast spells feature.
- Losing your spellbook has no effect on it. (That's true for cantrips as well, but those are spells.)