1. The $15 is for all parts and material, stuff like screw driver, etc. can be used outside of cost.
Unless you have some of the stuff you need (and are allowed to use it), you won't get far on $15.
I like the idea about the Styrofoam and spike (as wheels are costing a lot within our budget), do you have an illustration by any chance of how it should look based on your comment.
No sketch, but it shouldn't be that hard to imagine a wheel of spikes?
A hole saw and some MDF or plywood would be good for making hubs to hold the spikes (which could be any strong metal rod really).
Moving it would be by changing the CoG to just make it tip over one set of nails (each side should be angled alike and hit the steps at roughly the same time). When this has happened, the CoG is immediately adjusted for the next step.
Depending on size and step height/width, the egg in its cushion might be too little weight, so perhaps a good slab or iron (or lead if you have it and aren't scared that it kills you
) should be added.
A couple of servos might be good for juggling the "CoG adjuster" around, but they'll cost you more than $15, so I really cannot see it happen that way (assuming that you have to buy parts for the entire contraption).
If the egg is to be fetched and put down as well... Perhaps elastic bands would do for "motor power" (twisting them ahead of the run), as the amount of work is somewhat limited and this would save you a lot.
Steam engine and a tea light perhaps
Shooting a heavy weight over the stairs by a strong spring and with a string attached. The string can then be reeled in, climbing the stairs in the process (I don't see this breaking the rules, as it deployed the weight itself).
A spring from an old style alarm clock or a similar setup from a wind up toy car for locomotion.
If you're allowed to use "trash" to sweeten the $15 budget, you can go nuts, but if not, you really have to test any idea you and your team can come up with.
Just unsorted and off the top of my weary head, hoping it gets you into lateral thinking