So, there are a few layers of questions I’m seeing here.
1- Should you be more involved in parenting this child? It sounds like you’re feeling very left out of all decision making, and your wife is refusing to discuss it, which is not a particularly healthy situation. You need to find a way to discuss this with her, whether it’s via self-help book (sounds like you’re reading!) or professional therapy; you can’t have major off-limits topics that affect your daily life like this.
2- Is your stepson manipulating his mom? Maybe, but not maliciously. He’s trying to live his life, and he *does* need support. Kids manipulate constantly, to the point where calling it “manipulate” is a dark and ugly word for “getting what they need using the limited methods they have as a powerless child”.
3- Should ADHD kids be expected to do things? Yeah, they should. BUT. What he needs to do isn’t getting communicated in a time/place/method that works for him. Your wife shouldn’t be asking five times then doing it herself; he should be able to do tasks. HOWEVER, the solution is not “He’s a bad, lazy kid that we should be punishing, even if it’s a ‘natural consequence’ that leaves him in the lurch” or ANYTHING in that realm. The solution is on the communication (the adult’s) end. She needs to find ways to get his attention, times that work better than others, schedules that make things more routine, methods that make tasks less odious, and pretty much every other tip on the ADDitude site. This is a lot more work than constant nagging, but I promise you the kid isn’t enjoying the current setup where he’s made to feel like a failure on a regular basis, especially as a teenager.
Also, to caution against an idea further up on this thread that might be misinterpreted (I like it as written, but I can see NT people not getting it): Yes, ADHD people need to do things RIGHT THIS SECOND if they’re going to get done (hi, time blindness!) but making a 13-year-old drop what he’s doing do to an adult’s bidding RIGHT THE SECOND YOU SAY is not a recipe for home happiness.
As a middle school teacher and a person with ADHD: what will probably work best is to sit down with a blank schedule once a week (Sunday?) and go through everything the kid will need/want to do, and see how that fits. Make sure the kid gets a solid amount of say in how this goes; will homework be early/late, when will downtime occur, etc. Then, your job is to enforce the schedule that everyone agreed on. That might involve reminders, helping the kid, setting a lockout timer on the internet/TV/Phone, etc. etc. Tell the kid in advance of ALL plans you have in this regard (don’t surprise them with an internet lockout! They should know it’s coming!) If you or the kid don’t like how things are going, you can adjust the next week, and you’re flexible within reason, but the #1 goal is to help them learn methods and skills that work for them to *make that schedule happen* on their own, and knowing that they’ll need support (“let’s clean your room together on Tuesday at 3:30 after the internet auto-shuts-off” might be how this starts, and maybe by age 17, you’ll just need to get eye contact and say “It’s 3:30! What is on your schedule for this time?” or even, in heaven, his phone will do that for him).