I'm just shy of your age group (39 in October), but I take zero days off.
Attached FileI certainly don't endorse this for most people, though. I worked up from running 4 days to 5 to 6 per week over years, then just happened to discover that if I did a short, very low effort run on that "off" day I actually felt better the next day than if I had skipped that little recovery run. Your mileage, literally, may vary.
It sounds like you have a good plan. Make sure you get to a real running store to be fitted for some shoes. Try out a bunch and go with whatever feels the most comfortable.
I don't know of anyone who takes a full week off after 3 weeks of running, but many will take a "down" week to recover - maybe running 70-80% of their usual mileage. These people are mostly avid runners training for specific events, though. If you're running 3x days per week and not more than 3-4 miles per day you may not need to even take that down time at all - it's not a lot of stress and you're giving yourself plenty of recovery between each run.
If you aren't doing any running now, jumping straight to those three runs may be too much. Take it gradually. Maybe start with just a mile or 2 depending on your capabilities and add a bit of distance every week or two. I'd even skip the workout until you're running the miles per day you want for a couple weeks and THEN change an existing run to the workout - this way you're not adding both mileage and intensity (speed) at the same time. You level off, then increase either mileage or intensity - not both at once.
There's nothing wrong with running on a treadmill, but why not outside? If you have a decent hill nearby, hill sprints are a great workout. Less stress on the body than running at the same effort on flats because the ground is "rising up" to meet you. Sprint up a hill until you can't any longer, then walk/jog to the bottom and repeat.
Definitely make sure you up your protein a bit, and maybe collagen as well (beef bone broth is good for this) to help give your body what it needs to build up the strength in your legs/joints.