Supposedly the strong forearms can increase punching power, because thats the muscle which has to support the punch. I don't really know how true that is, but if I'm not mistaken thats what bruce lee thought.
3 minute rounds are too long and will throw off your pace/workrate. Start with 2 minutes and work on 3 minutes when you're getting ready to go open class. 5 rounds of shadowboxing is too many for a warmup. Do 2 or 3 light rounds of shadowboxing, with the first round being more of a hop-around-and-dance to get warmed up. 6 rounds on the heavybag is way too many. Start with 3, and alternate them with the 3 rounds on the double-end bag. Get rid of the tire-hopping. Jump rope for 20-30 minutes after your workout to warmdown and get some extra cardio/coordination training. If you want, add some running 3-5x per week in the morning or at night after training. It should go without saying that you do this in the gym with your trainer for the first few weeks/months before you're ready to do it on your own. If I had my way, you'd just cut out the double-end bag and heavy bag and work exclusively on focus mitts when you're in the gym. Other boxers should be able to hold for you and you can hold for them as well. Don't suck at it.
Warmup I don't like static stretching before boxing. Do a 2-3 rounds of just light shadowboxing/dancing around. Incorporate some dynamic stretching if you feel like it. You can do more intense shadowboxing later if you want to, for now it's just a warmup. Sparring/Drills Get your sparring in first thing. If it's not a sparring day (or if you don't spar yet), hopefully your trainer has some new drills/techniques to show you guys, and this should take up the beginning of the class so you can work on it later. The drills might be just shadowboxing, or maybe sparring drills. Sparring could be 1-4 rounds, drills could be 20 minutes to 12 rounds. Heavy Bag/Double-End Bag If you have both, shoot for 3 rounds on each. You can alternate them if you want. If you only have a heavy bag, stick with 3 rounds and work up to 6 or so. You're not just wailing away on the bag either. Your coach should have drills for you to go through. We usually end up doing the same one or two combinations for the whole round, often the whole session as well. Focus Mitts If possible, get this in first thing after sparring/drills. Pair up with somebody and alternate rounds working on specific things. Hopefully you can get a few rounds with your trainer or someone more experienced than you. This can be 3-6 rounds. It's time-consuming. Finisher: Something that really hurts. Burpees, sledgehammer, tire-hops, floorwork, basically whatever you need to completely obliterate yourself. I recommend rounds of burpees mixed with shadowboxing. You can pass out, and it's pretty awesome how tired/mentally drained you can get. Warmdown: Some combination of jumprope and speed bag. Go by minutes, not rounds. You can substitute the jumprope for the speedbag, never the other way around. That's how it usually works out for me. Different gyms have different ideas though. One coach I had put us through a 15x3minute round circuit: (Order varies depending on where you start.) Heavy Bag Uppercut Bag Focus Mitts Jump Rope Jump Rope Medicine Ball Drills Medicine Ball Drills Double-End Bag Shadowbox Double-End Bag Uppercut Bag Speed Bag Speed Bag Shadowbox Heavy Bag This was after 20-30 minutes of basic warmup/shadowboxing combination drills. The more advanced guys would get there early to spar and then do their own workout during the basics part. Regardless of what you do in the gym, you should be going 3-6 times a week and doing running on your own time if you plan to compete.
Punching power comes from the legs, hips, back and a little bit from the arms (triceps). Why would you train to strengthen the end of the chain when there are many other more powerful links you can put your time and energy into strengthening? If you can provide links to studies proving stronger forearms = more powerful puncher please do so. For your own sake you really need to stop taking Bruce Lee bull**** as gospel.
You can't go wrong with sports specific exercises. What you posted will increase your skills as well as your overall conditioning. Looks fine for now. There is no reason to isolate any body part, muscles that damage will super-compensate. And sports specific exercise always damages the right muscles. Your focus should be on acquiring skills at this point anyway.