The answer to question 2) only you may give. You're the one who's worked there and dealt with those people. However, let's analyze the situation as a whole:
You work in an organization which by your own words lacks ethics and has management issues. You were upset enough about the things you experienced to make you want to leave, and found a good job elsewhere.
You have a pretty good offer on the table, but your ethically compromised bosses have returned with a counter offer: a boatload of money, a promotion, and the opportunity to telecommute! Sounds fantastic, congratulations! Right?
Too Good To be True
These guys could have given you a raise and listened to your concern long before now. A sudden $20K pay bump, promotion, etc. smells of desperation to me, and that won't work out well for you in the long term.
Why? Because no one is irreplaceable, and your lack of loyalty will never be forgotten. They might need you desperately now, and are willing to bend over backwards to secure your cooperation, but the second whatever hurdles they're facing are behind them you will be shown to the door, and make no mistake about it.
We all want to be that person who joins a poorly managed organization and somehow turns the whole thing around into a wildly successful enterprise. That happens in real life, but when it does it's because someone's fronting a lot of money and changing out most of the staff, never-mind the internal processes.
You are deluding yourself if you think that you will somehow take up a position of leadership and lead them all into a bright new future. This would have happened before now if the people in charge wished it. They are the ones driving the culture of the company and making the final decisions. The fact that they have not taken any steps until you were half way out the door tells me that this is not actually important to them, but simply something to lure you back in.
Do you honestly think you will have their full support in completely overhauling the way that the entire company works? To push major, disruptive changes into many departments? Hire the staff to make those processes a reality? Tell other managers how to run their teams? Dismiss or punish uncooperative staff? How about develop and implement the training, all while developing products, and pushing features out the door? Right.
If this company is ethically compromised, and poorly led, and knowing that statistically this is not likely to change, why would you stick around for more of the same behavior which drove you to quit in the first place? Especially knowing that you will be painting a target on your back for later retaliation as an over-paid, disloyal employee?
I'd take that other job offer and never look back.