Somehow the feature that tells me of software updates has become corrupted, and so these things aren't happening. How do I fix this?

Version 15 wasn't long-term-support: I think the support ended in January of this year! You need to get version 17, which will keep you in updates until 2019.

Well, I'm running 'small' hardware. Would like to try to avoid the later and bigger releases. Just want to get the available fixes for this one, and leave it as it is...

What are your hardware specs?

Unless you are doing something really resource hungry, such as editing video, the biggest variable in resource usage is generally the desktop environment itself. Generally, what distinguishes so-called light-weight distros is the use of light-weight desktops.

If you want to stick with Mint, you could install the current release, then install and use a light-weight desktop such as LXDE or even a window manager such as Fluxbox. I set up a computer with 1GB RAM like that for my son and the performance was acceptable when he was logged in to LXDE.

You would leave the default desktop in place, just not use it.

What I have is two identical Acer Nettops with 1.8 Mhz Atom processors, 2 gig of ram. This one has the Mint 15 distro, though I've forgotten which environment is on it. The other machine, which my wife uses, has Mint 16, and I made a point on that one to use the LXDE environment, based on an earlier cycle of trying to figure out why these machines were kinda doggy. It's truthfully not faster than this Mint 15 unit, in spite of the lighter environment. Some of the problem was mitigated by moving away from firefox to opera, but these machines are not what most people would be happy with, though they're O.K. for email and casual browsing, if you're not in a hurry...

I'm REALLY trying to avoid upgrading, as I see no value in this for these modest machines. It's a pain in the rear, for what might be a downgrade in performance. Linux, like all software, grows with time. One has to decide where the limits are on an individual basis.

This computer has a 32-bit processor and 1GB of RAM. I use it for reviewing distros, and Mint 17 Mate ran perfectly on it. I can't remember much difference difference between versions 15 and 17.

Neither Mint 15 nor Mint 16 are supported any longer. If you want Mint, your options are Mint 13 (support until April, 2017) or 17 until April, 2019.

That should be plenty for most contemporary Linux distros, though you might want a lighter-weight desktop. I'm running Slackware --current on a computer with similar specs and it works just fine (it's got an Intel CPU, though, not AMD). I do use E17 for my desktop, because I really like E17. It and Fluxbox are my favorite GUI interfaces.