I've done this many times on other bikes. Usually, I loosen a very long screw (pic 1), and sometimes I also loosen the other part (pic 2). But this time, no matter what I do, the handlebars on this bike won’t move up, down, left, or right. How can I adjust them?
When you loosen the long bolt at the top of the stem - where you've labelled it with the 1 - does it ever become free to move up and down even though you can't remove it? I'm guessing it doesn't, meaning the internal parts of the stem and headset are seized together.
What you have is a quill stem, most likely with a wedge at the bottom that the long bolt pulls up to wedge the stem tightly and hold it in place. If the wedge loosens, the long bolt is able to move up and down with the wedge still attached. If the wedge is seized in place, the bolt won't move up and down - you can only unscrew the bolt entirely.
Given how rusted the headset bolts at the base of the stem are, there's a really good chance that the inside is just as bad, if not worse.
If the wedge is seized, there's a good chance that it can be freed by loosening the bolt so it extends well above the stem and then hitting it with a hammer or wrench. You don't want to hit it so hard that you bend the bolt, though. So no 2 kg sledgehammers.
If that doesn't break the wedge free, you'll probably need some penetrating oil* to help. Getting the oil down to the wedge will be a bit difficult. Sending a good bit down the bolt hole might work, though it'll probably waste a lot of the oil and create a bit of a mess once you get the stem free. If the fork is open at the bottom of the steer tube (between the legs of the fork) you can turn the bike over and apply to oil through the opening, where it should run down to the wedge easily.
Let the oil soak for a while and it should break free rather easily.
But that's just the wedge.
The rest of that headset might be seized too, from that picture, which could also prevent you from raising the stem. The stem could very well be seized to the headset, for example. In which case, the same basic techniques can be applied - see if you can work it free with mild force and/or impact, and if not soak it in penetrating oil, give the oil time to work, and try again.
Finally, once you get the stem free, there's a good chance it's not long enough to raise the bars much further. That stem already extends a good long way above the headset, so it might not be long enough for you to safely raise the bars. In that case, you're going to need a longer stem.
* - and no, "basic" WD-40 is not a penetrating oil, although there is a WD-40 branded penetrating oil. But I'd bet it's nowhere near as good as Kroil...
Loosen the top bolt a few turns, then give it a firm tap downward to release the stuck quill wedge, after that the bars should move.
