The real issues here are two:

1 - How big the existing cut out is

2 - What shape the countertop is in the area currently covered up by
the sink where it's resting on the countertop. By this, I mean if it's
in good shape, then the only limitation is the hole cutout. If for
some reason the countertop under the existin lip doesn't look good and
would show if the new sink lip is smaller, then that becomes a
limitation.

You should be able to find online documents/drawings for any of the
sinks you are interested in that will show the dimensions of the new
sink.

The new sink will come with whatever is needed to install it. The ones
I've worked with have been all one piece. Generally, a light sink,
like stainless steel, will have clamps that secure it from the bottom.
A heavy cast iron sink will not, they just rely on the weight plus
caulk to hold it in place.