Spin class--clips vs. cages?

Use the cages, unless you have an extra pair of shoes around.  I use some 13+ yo shoes that I had lying around.  

I will say that I love spinning in cycling shoes, and I do think it makes a difference, but I would hardly call it worth the trouble.  I probably wouldn't bother with spin class if I had to use cages.

I would get an inexpensive pair with cleats that fit the spin bike.  You just can't get that upward pull as well when you're in the cages versus clips.  Over the course of several months, that little bit you're missing can start training your legs to focus too much on the push and not enough on the pull.  I've also seen many students suffer from biomechanical issues (i.e. foot and knee pain) over time from misalignment caused by the fact that that foot is not really connected enough to the pedal.

If you were spinning once a month, I'd say no worries.  But once or twice per week, I'd invest in an additional pair of shoes with the spin cleats.

I use Crank Brothers on my bikes, but of course the spin class bikes are SPD.

I started out in gym shoes with the cages, but hated that.

Then I switched to my bike shoes jammed into the cages. Better with the stiffer soles, but still lousy.

Then I remembered I had a pair of old slightly-too-big bike shoes in the back of my closet. I bought SPD cleats for them. I'm happy with this even though my feet do move around a little inside the shoes.

I'd suggest getting cheap, or used, shoes that fit "okay," and putting SPD cleats on them. They don't have to be as nice as you would buy for regular riding, but I do recommend that you do one better than me and get shoes that fit. :-)

No harm at all in using the cages. 

As to the above about focusing too much on the push and not enough on the pull?  That's exactly what you want to do.  Moving a bike forward is all about applying pressure down on the pedals.  Not about actively pulling up on them.  Beyond unweighting the pedal, any active pulling is largely a waste of energy.