Convex Mirrors

[Soaring Crane]

I know that magnification of a single convex mirror is always positive (that is, greater than 0) based on f<0, but does the size of the magnification depend on the magnitude of the object distance?

[Gokul43201]

The simplest way to answer this question (yourself) is to draw the ray diagrams for a convex mirror putting the same (sized) object at two different positions, and see if they lead to different image sizes.

The other way to answer this question is to derive the relation for the magnification from the formula :
1/u + 1/v = 1/f. This isn't terribly hard, if you remember that the magnification, M = -v/u

[Soaring Crane]

Can m for a convex mirror ever be equal to 1 aside from being greater than 1?

[GCT]

Can m for a convex mirror ever be equal to 1 aside from being greater than 1?

nope, as you said in the original post