I've read that adding cork (from wine bottles) absorbs the fat. I'm not sure if this works or not.
Are there any other ways to remove excess fat without standing there and spooning out or refrigerating and then removing the solidified fat?
Soup – How to remove excess fat from stews or soups without refrigerating
fatssoupstews
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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.
For the very best tasting fries, onion rings and battered fish are fried in fat made from rendered beef fat. When I was a cook we rendered down a thousand pounds of beef fat a week, it took days to do. But it made the very best tasting savory deep fried foods. The burning temperature is lowish, so food needs to be cooked at 325 and changed more often. It's not good for things like doughnuts.
I'm not a fan of Canola, I find it often has a taste which I don't like. I don't use peanut mostly due to habit, too many people I've run into with allergies. I can't find rendered beef fat and I'm not going to render it at home.
At home I use a mixture of Sunflower and lard, about 1 part to 3 parts. The sunflower oil makes it easier to handle when filtering through a very fine mesh sieve (more liquid at a lower temperature) after use, then I put it in an old olive oil can which I refrigerate between uses. I leave out overnight before I use it then warm in hot water to pour it out. I don't usually use more then once a month. The more crumbs and batter bits you filter out the longer the fat will remain good. So filter, and don't pour the last bit into the can.
Refrigerated the oil mix doesn't go bad before it becomes no good to use. I get well over a years use. I love sunflower oil, it has almost no taste at all and a high smoke temp. It is pricey.
Best Answer
If you want to be really lazy about it, just get yourself a fat separator. Pour in the soup, the fat will rise to the top, and you can do what you want with it (i.e. dump it).
If you're reading this in an emergency, you can do this with just a strainer. You'll get better results if you chill the strainer before each skim, i.e. by rinsing it with very cold water. The fat will tend to congeal on the cold strainer the same way it congeals when it's actually chilled.
I've also heard that the fat will cling to certain leafy vegetables, like lettuce. If you have a lettuce head kicking around, try peeling off a leaf and dusting the top of the pan with it.