would you mind checking for me if my constructions of the following sentences for a model is correct? Thanks!
1) The model has exactly one member ∃x∀y (=(x,y))
2) The model has at least 2 members
∃x∃y (¬=(x,y))
3) The model has no more than 3 members
∃x∀y (=(x,y)) v ∃x∃y∀z (¬=(x,y)&(=(z,x)v =(z,y))) v ∃x∃y∃z∀w (¬=(x,y)&¬=(x,z)¬=(y,z) & (=(w,x)v =(w,y)v =(w,z)))
4) The model is infinite
∀x∃y (<(x,y))
For (3), your sentence is correct but far more complicated than it needs to be - I would have used a sentence that says "there are not four objects which are all different".
For (4), I assume you have some standard meaning assigned to $<$? First-order logic usually doesn't include that as an "assumed" symbol like $=$, but it's possible that your class does include it.
(1) and (2) are correct.