First off, excellent choice on taking charge of your physical fitness. 6-9 months can seem daunting, but really if you dial your nutrition in coupled with an effective training program you will see remarkable results. I'll try to chop up your question a bit:
But I'd always heard (in my complete ignorance) that you should not do
weight exercises every day, but should take breaks between days. Would
that apply even to these basic exercises?
There is a stress-adaptation cycle that's happening. Exercise that's just ever-so-slightly more than you can handle will break your body tissues down a bit (stress) and then then your body will repair the damage plus add a bit of margin on to be safe (adaptation). Adaptation is specific, it's small, but it's the key to all physical fitness.
So specifically to answer your question you'd initially need to workout every other day, probably just because you'll be sore. If you stayed on that routine you linked, your body will catch up, and it will never need to adapt because you're not challenging it anymore. And since you don't need to adapt, you don't need any rest in between. It's the same reason you can walk x distance every day, day after day. It's a stress your body can handle.
Generally, what problems might occur with these exercises? Should I
watch out for strains from the dumbbells?
The biggest problem is that the "routine" they put together is woefully ineffective. I did some poking around on Rick Bradley's QuickFit program (LLC). You can believe me or not right now, but in 6-9 months when you're wondering why any progress you've made is barely detectable at best, please come back to this answer and read the next paragraph.
You need a program. Not a "routine", which is some marketed garbage spewed out by "fitness industry experts" to make money. You need to embrace the fact that your body was designed to do amazing things and you really can, quite rapidly, become incredibly fit. It doesn't take a ton of time but the things it does require are not up for modification. Compound lifts, good nutrition, and progressive loading.
I'm roughly your same age (actually older), I have a family with small children, a full time job, and the regular trappings of life. I went through this path myself, I've helped others, and I promise if you do it you'll be kicking yourself for not doing it ten years earlier.
This doesn't need to be an all day affair. Good nutrition and solid training will have you exercising three days a week for maybe an hour tops: that's the equivalent of watching a long-ish movie every week. Except that one is mindless entertainment and the other makes you stronger, healthier, and increases the enjoyment of every aspect of your life.