Compressor does draw higher than rated for a very amount of time so wiring sized for 15 amps is sufficient (14AWG) most contactors can handle above 20amps and are designed for motor loads . So any contactor is sufficient. I used a 30 amp one which is overkill but no problem oversizing. I would check for UL listing. Packard is name brand i commonly use at work and is economical and reliable. . It is unlikely that a failure would cause any issues other thank your AC not turning on in case of failure. They do last a long time considering how small amperage draw residential equipment draws. As long as everything is grounded i dont see any issues . If you are worried about electrical safety i would consult with an electrician or purchase a manufactured relay or built by someone knowledgeable in electrical.
As far as size for box, i used a 6x6 outdoor box to waterproof components and have plenty of space for wiring.