You can do it together, but it might end up causing you issues because depending on your immigration office, they might be handled by different departments/counters. Which means they won't let you apply for your extension before you go and get the stamp moved. Which could take hours or the entire day. That could mean you can't apply for the extension that day because you had to get the stamp moved first.
The officers at the airport can also get a little annoyed if you try leaving Thailand with an extension and a new passport, because they know you should have gone and gotten it moved already.
So technically, you can do it together. But to avoid any potential issues it's best to do it as soon as possible.
Yes, print it. You don't want any mistakes like being stamped in visa exempt, or the airline not letting you board because you can't show them a visa.
Your timing is fine. The date on the visa is when the visa expires. That's completely separate from your permission to stay in Thailand. When you enter, your visa will be gone, used, invalid. But you'll receive a permission to stay stamp that will be for 90-days based on your non-O visa.
You're just planning on going this week, or your entry stamp expires this week? You need at least 15 days left on your stamp to apply for DTV extension.
You're not actually on overstay. That's just the only mechanism they have for fining you for failing to cancel your extension. The only thing you should expect is a fine, which is 500 baht per day since you stopped studying, to a maximum of 20,000 baht. There shouldn't be any other issues now or in the future.
You should have qualified for ED Plus after you finished studying though.
You would need to go to your immigration office to find out their specific requirements. Most of the paperwork will be the same paperwork that your spouse used to get their work visa, and their employer will also have to approve and provide the required paperwork.
But I don't believe immigration recognizes domestic partnerships.
It's not your friend's application, it's your application. And you don't need any visa at all for a 1 month course. You are lucky they even gave you METV.