This is a rather hard one to answer but I will do my best. Let me start first with answering your question on what would be the best way to teach 3rd world countries to repair the devices they would be getting. This is challenging for a couple different reasons but the main reason is because they don't have access to knowledge like you or I do. If we want to know something we generally can hop online and start researching.
That's not the case for them. Even people here in the states without internet can simply go to McDonald's or Starbucks and Bam free WiFi. That's not the case in 3rd world countries. So the best solution I can think of is a little bit more expensive but would work wonders! The answer would be to create or find guides right here on ifixit, and print them off. Then simply package them with the device being sent and users receiving your donation would get the device and a guide complete with how to maintain and repair the device. Your next challenge however would be, "how do they pay for or even go about buying replacement parts"? It might be easier to simply print off a return shipping label which would allow them to return the device to be fixed free of charge. However this can get VERY expensive as you will have to pay for shipping labels that may never be used or are even used for shipping other items besides the device. In addition you would then have to pay the repair price. Leaving you broke, so perhaps not the best idea unless you could partner with another company that already does repairs.
Moving on I would like to answer your next question of, "Are there any companies that donate to third world countries?". I tried finding some charities that have a goal of donating devices. I cannot find any. More likely you will find companies that accept money and in return donate to different areas like medical research or food etc. They aren't worried about giving 3rd world countries a mac book pro or iPhone 7.
However some companies like Dell, HP, and other computer manufacturers have been known to release campaigns where they donate computers or other accessories based on their sales for that given campaign. If you are serious about your efforts it wouldn't hurt to contact Dell or another company and see if some kind of arrangement can be made in order to not only donate devices but maybe fix them up?
Getting Dell to partner on that would save loads of money and headaches! Think about it, you send the devices and dell repairs them. Im sure more details like money would need worked out but they are been very charitable in the past. Perhaps they will get on board??
As for the E-Waste you answer is both yes and no. While repairing old devices and then donating them does stop them from getting thrown out at that time, eventually they will be abandoned, trashed, or just flat out resold. Until we can find materials that break down easier in the environment, donating old computers simply transfers where the e-waste lands. To solve e-waste we need to make our devices out of more earth friendly materials. HOWEVER your charity would help circumvent how much e-waste there is as at one time.