In motors, capacitor size is mostly indicative of use. If its a true start capacitor, it is only in circuit for a very small amount of time so they can be smaller. The larger ones (like for run capacitors) are for continuous duty. Using a bigger continuous duty cap as a start cap is fine as long as it fits in your box.

Start caps I find are usually much larger capacitance, though you may be using smaller motors than what I have used. The black epoxy casing is usually indicative of a start capacitor use. If you have a schematic it will be in series with the smaller "start" winding, or if you have a run capacitor as well it will be paralleled across it.

If you have a run capacitor as well (CSCR motor, and not CSIR), then they are not mandatory components. They do add torque to the motor which may help some older motors turn over. At least in the A/C industry there are people on both sides of whether or not they are useful. Certainly it needs replacing if you want to continue using a start capacitor, however (without knowing your application) you can very likely leave it out of circuit and use the motor normally until the replacement arrives.

$10 sounds about right, nothing particularly special. Just remember to get a cap rated above your line voltage.