Hello,
Welcome to the forums

I'll try to answer what I can, hopefully somebody else will be able to inform you when it comes to texturing and modelling.
For tutorials there are two places I usually go to,
the3rdage.net and
here! The tutorials here are in german so you'll need a translator but they touch on quite advanced topics.
For opening mod files you can use something like FinalBIG (or my version with updated functionality,
FinalBIGv2). For editing the files any text editor will do, something like Notepad++. I use Sublime Text 3.
What is the best approach for editing unit stats and balancing them without breaking gameplay? I have read through some of the INI files; but I am worried about introducing errors. Are there common mistakes I should avoid?
Always a hard question, specifically for Edain most things are standardized so if you want to change something you just have to look at the numbers and see how they compare to other "tiers". If you have specific examples of balancing you have in mind and what you fear that will break that would be handy.
For errors, the game will usually crash pretty loudly if you make a mistake. The recommendation is to never spend too long without launching the game. Do some changes, launch the game to check for any errors, do some more, launch the game, etc... Otherwise you may code entire units and then get stuck with a cryptic error message.
For testing, you have two approaches. You can either make copies of .big files with higher name priority, INI.big -> !!!INI.big, and edit those. The game will read the file with higher name priority last making it override other data. The other possibility is to use the -mod technique, which allows you to edit files unpacked instead of packed in .big files. Some tutorials on that:
Finally, my advice when it comes to modding BFME is that there is a lot to learn so don't launch straight into a grandiose project. Start with smaller changes, like adding one hero's ability to another or changing a texture.
Feel free to ask as many questions as you want here