The simple answer is no, you cannot convert monounsaturated/polyunsaturated fats into saturated fats through cooking alone.
Before I can even begin to answer in detail, I have to start by pointing out that "saturated" and "unsaturated" fat is already an oversimplification. These are very rough classifications of fats and the chemical reality is far more nuanced. I'd encourage you read the Wikipedia entry on Fatty acid for a relatively basic explanation.
Fatty acids are actually classified, chemically, in three different ways:
The existence of double bonds (CH=CH) somewhere in the molecular structure. Fatty acids with double bounds are unsaturated. Fatty acids with single bonds only are saturated.
The chain length (number of Carbon-Hydrogen groups). Another distinguishing characteristic of saturated fatty acids is that they are generally (maybe always) long-chain. The reverse is not necessarily true, however; not all long-chain fatty acids are saturated.
The configuration, either cis or trans. Trans fatty acids are not actually a different type of fat in the same sense as saturated vs. unsaturated; they are actually unsaturated fatty acids, just in a different configuration.
Although certain studies suggest that it is possible to create TFAs from edible oils with prolonged heat, it is also extremely difficult, to the extent that it's nearly impossible to do in meaningful quantities by accident. I will refer you to this study of heat-induced cis/trans isomerization which says that after 8 hours of heating at 180° C, the researchers found up to 6.5 mg of trans isomers per 1.0 g of oil, which comes out to a grand total of 0.65% by mass. This is practically nil as far as a home cook is concerned - these quantities only matter if you're doing commercial processing of vegetable oils, or maybe if you're using the oil for deep-frying and you reuse it dozens of times (far beyond what any experienced fry cook would recommend).
The study also says that edible oils (i.e. the ones you cook with) experienced less isomerization than other kinds. So really, the amount of isomerization you're going to get (conversion of the "good" cis isomers to "bad" trans isomers) is minuscule and simply insignificant as far as mainstream baking or frying applications are concerned. So forget about trans fats.
Can you create saturated fats? That would effectively mean breaking down the double bonds into single bonds. In order to do that, you need to add hydrogen (that's why saturated fatty acids are "saturated" - more hydrogen).
There's a name for this process, which you actually stumbled upon in your question. It's called Hydrogenation. It's adding hydrogen atoms to an unsaturated (double-bonded) fatty acid.
Hydrogenation requires a substrate (involving benzene or some other hydrocarbon), a hydrogen source (that's pure, dangerous, H2 gas), and a catalyst (heavy metal). My guess is that your kitchen has none of those things, unless you're cooking in a chemistry lab. So there is simply no chance for you to accidentally hydrogenate your oils.
What you really need to be more worried about with oils (unsaturated fats) is lipid peroxidation. That's the oxidative breakdown and, eventually, rancidity of fats, and polyunsaturated fats are particularly prone to this. Heat is a catalyst for peroxidation, so if you "burn" your oil (or other fats), you may end up creating the same sorts of free radicals normally associated with rancidity due to improper storage. The long-term effect of these free radicals is not firmly established but the consensus seems to be that they aren't good for you in the long term (cancer risk and so on).
So don't worry about converting your oils when you cook with them. It's practically impossible. You should be more concerned with overheating them or letting them go rancid in storage.
Gelatin works by creating a very fine mesh of proteins, between which the (hidrophilic) liquid gets trapped.
A mixture of fat and water isn't a liquid. It can be either a rough two-phase mixture, with visible fat droplets swimming around in the water, or it can be an emulsion, with invisibly small fat droplets dispersed through the water. Emulsions appear smooth, e.g. milk.
When you try to use gelatine on the mixture, two things can happen. In an emulsion, the fat droplets can be smaller than the protein mesh. Then they get trapped as well as the water, and the result is gelled emulsion, just like you'd expect.
But if you have big globules of fat, they can't be trapped between the proteins. So it looks like they got squeezed out of the mesh and bubbled up to the surface, if it hadn't been there all along. On the surface, gelatine molecules couldn't find each other between globules of fat to crosslink, or maybe the motion of the globules broke any nascent links in the small amount of water between them, so it couldn't gel. So the conclusion is: you can't set soup/stew with gelatine. (I know that concentrated stock hardens from its own gelatine, but it has the fat filtered out first).
On the practical side, if you want to have thick stew juice, you have to use something else than gelatine. It is probably simplest to create an olive oil based roux with the stew juice. Or you can gel with xanthan. Both will result in a "juice" which is creamy, instead of solid wobbly blocks of juice, with xanthan retaining the authentic taste and a roux-based sauce being a novel variation.
If it is real jelly you are after, you'll have to emulsify first and use gelatine after that. Xanthan itself should emulsify well, or you can probably use lecithine. AFAIK, there is no problem to use lecithine and gelatine together, but somebody with more experience in gelling agents should confirm that. In any case, emulsifying will lead to your liquid getting quite opaque. This is inavoidable, as it has to do with the way the collection of individual micro fat droplets break and reflect light rays.
Best Answer
From experience with gelatinous stock from pig bits, I've found that gelatine can seem fatty, when really it's not - and gelatine is good for you in ways that fat is not.
I think your tall, narrow container idea is a good one, but first get it up to liquid temperature, and then keep it at that temperature for as long as it takes. You'll want to avoid convention currents, so take it off any direct heat. It might be enough to wrap it in a towel, or maybe you'll need to leave it in an airing cupboard or even a low oven.
Given long enough, even the emulsified component should separate out - although I can't guess whether that would be hours, days or weeks!
This does mean leaving the stock at a bacteriologically unsafe temperature for a long period - so the stock you test in this way should not be eaten. Do it on a small sample.