I get the 'no budget' thing. I've worked with schools, churches and other organizations without a proper sound system and have been able to get some good basic mixes. Similarly, there might be some relatively easy fixes for these issues. Some of the 'popping' sounds might be removed by simply buying new cables for your microphones and speakers and checking all of the cables (and mixing board channels and effects... yes they can go bad) in the signal path. You might also be picking up 'handling noise' from your microphones. Some mics are worse than others about this, and you should get some time alone with the system and see if you can reproduce the problem to know for sure.
For the balance problems, there are other things to try. First, if you have any influence at all with the performing groups, search around the web about microphone technique (here's one... http://www.voicecouncil.com/mic-technique-for-singers-101/ and another one... https://music.tutsplus.com/tutorials/microphone-technique-for-vocalists--audio-2469) and try to get your groups to use some of the basic handling and presence techniques in order to get a better sound. For those shyer performers, you may need to give them a nudge in the channel gain department, but really if they are giving you a good, consistent signal to work with, it should be fine.
I would also caution against using equalization (EQ) to try and solve a bad signal problem. As you noted, things start to get strange very quickly when you over-use the equalization on a channel. And to me, "over-use" means anything more than +-6db on any frequency band. Other people may have other ideas, but the main thing is to get the best sound possible for the group you're working with. To that end, you should try researching some live sound engineering techniques. Most of them have to do with bands in clubs, but the same basic techniques apply (this is overkill, but a great basic introduction to live sound reinforcement: http://www.electrovoice.com/downloads.php?type=PA%20Bible). Probably the best idea when starting out is worry about the vocals first. They will probably be the primary focus of any mix and you might not even have to amplify any of the other instruments (especially if they have their own amplifiers). Good luck!