For example I doubt a moon could appear much bigger than the current one is without us becoming the moon.
Not quite.
This page claims that the Moon was 3-5 Earth radii away when it formed, or ~19 000 to ~30 000 km away. At this distance, its angular diameter would be between 10.5 and 6.4 degrees. Currently, the angular diameter of Luna is around half a degree. This is a pretty huge difference in angular diameter.
Such an arrangement may place the barycentre outside the Earth's surface, but it won't change the mass of the Moon- it's still be far, far less massive than Earth (thus, we wouldn't be
its moon, at least not by a definition of mass ratio).
Of course, if we're talking objects less massive than Earth, we can go all the way to significant fractions of Earth's mass. If Earth and Venus circled eachother with the separation between the two being equal to that between the Earth and Moon, it would have an apparent diameter of 1.8 degrees. If the separation was 50 000 km, it would appear nearly 14 degrees wide.
One novelty is that if a satellite is less dense than the primary, but less massive, you can have a 'moon'
bigger than the primary object. For instance, swap out Venus for an object of a similar mass, but a lower density- around that of Pluto (say this is an oceanic planet, or something). It would be nearly 16700 kilometres wide- at 384 000km, it would be nearly 2.48 degrees wide, and at 50 000 kilometres, nearly 19 degrees wide.
So it is possible, in theory, to have a 'moon' larger than it's parent planet- but given the physical traits involved, the proper circumstances might not occur in planetary formation.
Of course, note that one must also be weary of Roche radii and suchlike.
Here is an image I took of the Moon, with a badly altered version to show what it would look like if it orbited at 35 786 km (angular diameter of over 5.5 degrees);
