Is this normal?

It might be "normal" in the sense that your hardware isn't broken, but you'll very likely get a better result with better equipment.

Am I overthinking this and these "defects" are actually unimportant?

That depends on how much the photos and the scans mean to you. If the photos are fairly stable now and you feel like you'll be able to scan them again if you ever need a better result, then that obviously decreases the importance of getting the absolute best possible scan now. If the photos are starting to fall apart, or if you definitely don't want to go back and re-scan later, then it makes sense to do what it takes to get it right this time around.

Would getting a new scanner be better?

Yes. Based on what you said about the inconsistent illumination, and also the 1200dpi max resolution of the scanner, you can surely get better hardware without spending a ton. Just looking quickly at what Amazon offers, I see the Epson Perfection V550 has 6400dpi optical resolution, ability to scan negatives (I know that's not a priority for your collection), and software that'll help you restore faded colors, with a price around $160. If you want to spend less, there's a Canon CanoScan LIDE 400 with 4800dpi optical resolution and fewer features for around $90, and that'd still be a big improvement over what you've got. I don't mean to recommend these specifically -- just to give a sense of what's available at very reasonable prices. You can of course also spend more and get more.

Would a more expensive scanner be better than a cheap one or are they mostly the same in this regard?

See above. You can get a much better scanner than you currently have without spending a lot, but you can also get incrementally better hardware at higher prices. Does the difference in image quality and/or features between the $800 Epson V800 and the $1000 V850 justify a price difference that would more than pay for the V500, which has the same optical resolution as either of them? You have to read the specs, read the reviews, maybe visit a dealer, and ultimately decide for yourself. It's clearly worthwhile to somebody, else the V850 wouldn't survive in the market, but you very well might not be the target audience.

Should I be worried about color-calibrating my scanner?

Maybe. Are you going to print these photos? Do you want the prints to closely match the photos in their current state? Are you going to display the photos on several screens?

Color calibration of the scanner won't make a whole lot of difference if you don't also calibrate your monitor(s) and printer(s). Since your old photos have probably faded and will need some sort of color adjustment, you could argue that it's much more important to calibrate your monitor(s) and printer(s). Even if the scanner shifts the colors this way or that, you can fix it after scanning and then know what you'll get when you print. On the other hand, if you're going to go to the trouble of learning about color calibration and then calibrating your monitor and printer, it probably makes sense to calibrate your scanner too, so that the images you scan will look like the originals right off the bat.