It's great that you plan on having community managers available. However, I would fear that even though a mentor may start off being very willing to dedicate whatever time necessary to their mentee, over time, things could change (the mentor could take more time to respond to the mentee, the mentor may not have as much free time, etc.). Although a rating system may not be necessary, I do think these managers should check in with mentors (perhaps monthly) to make sure they are still able to contribute the time and effort necessary to taking care of their mentee(s).
Regarding the need to understand the cost for sending kits to students, have you thought about reaching out to corporate sponsors? This would lower the cost of acquiring items for the kits, and it may even align with companies' CSR goals. Ex: Reaching out to GE to donate lightbulbs or Duracell to donate batteries.
I think it's great that this is serving the needs of students in schools that may not have strong alumni networks. One question I have is how would you determine when a mentor is ready to take on another mentee/if they would want to? Another idea that might be worth considering is incorporating tips for coffee chats with mentors. E-mail writing is great, but students should also have some guidance on conduct/questions to ask when meeting professionals in person.