Rice is mostly made of starch. Starch is, in itself, a molecule made up of glucose components attached to each other. There are two types of starch: Amylose - it is a long straight chain of glucose - and amylopectin, which has a branchy and fuzzy structure. When you cook a rice which is rich in amylose, the grains stay separate. When you cook rice which is high in amylopectin, its starch molecules catch on each other and cause clumping. So the main factor is indeed the type of rice. While the amylopectin rich varieties in general are short grained and amylose rich ones are long grained, it can be that you accidentally picked a non-sticky short-grained rice.
Washing will cause less sticking. Normally, the starch in rice has to be released from the cells before it can stick. In a bag of rice, there are many cells which are broken mechanically during handling/transport, and their starch is free, clinging to the surface. If you wash it away first, you have less sticky grain so less clumping.
Now for soaking. Starches are packed very close in a grain. For gelation (that's when they cook and swell) you need both enough water and the right temp (70 degr. C). As heating is quicker than water penetration, presoaking makes things quicker. I guess soaking will help stickiness a bit, because there will be more molcules ready to swell in a short time. Plus, some of these will come undone from the grains and start swimming free around. This turns the water itself into a weak glue (so don' discard).
Using less water will help with stickiness. This will result in a bigger concentration of free starches in the water.
A slow simmering should also promote clumping slightly, as the starches will have more time to swell, move around, and hook to new starches.
All arguments above are the theoretical explanation for the direction in which the factors you mention are likely to influence clumping, given that the rice is always the same. In practice, their effect should be much smaller than choosing the correct type of rice. In fact, amylopectin rich rice types aren't soaked as often, because they don't need it - amylose is packed tighter. I don't know about washing habits, but it is cooked with less water, because it needs less. And any rice should be cooked slowly, a hot boil overcooks the outside and leaves the inside hard.
In the end, if your wife wants the rice she is familiar with, you must buy short-grain japonica rice. Else ask for a "sticky" indica, that's better than just eyeballing grain length.
The basic differences between white and brown rice is explained below (as written here):
An important first question to ask about all rice-and for that matter, most foods-is how much it has been processed. In the case of rice, processing usually involves milling and polishing. The outermost layer of rice, called the hull, is removed to make brown rice. Brown rice is rice with the whole kernel intact and the kernel is still surrounded by all layers of bran.
To produce white rice, the bran layers of the rice have to be milled off. Most of the rice germ is also removed during this abrasive grinding process. At this point in the process the rice is called milled, unpolished white rice. Finally, a wire brush machine is used to remove the aleurone layer that remains on the rice. This step is called polishing. As polishing is not an all-or-nothing process, semi-polished rice may still contain parts of its aleurone layer.
Additional information about brown rice (here):
[Brown rice] has a mild nutty flavor, and is chewier and more nutritious than white rice, but goes rancid more quickly because the germ—which is removed to make white rice—contains fats that can spoil. Any rice, including long-grain, short-grain, or sticky rice, may be eaten as brown rice.
So, because they are processed differently and have different parts of the grain intact or removed, your method of cooking is going to vary.
I have used this method of cooking brown basmati rice with pretty good success. It includes washing the rice a number of times prior to cooking and adding salt. This method advises adding oil after the cooking process, but I typically add it into my rice while it cooks, or even sauteeing the grains in oil then adding water to the pot to cook.
It is also suggested that leaving your brown rice to soak for 15-20 minutes (and sometimes even longer) can be beneficial as well.
All of that aside: if you're looking to stick with a flavor and texture similar to that of white rice then brown rice may not be your thing. I know that as a jasmine rice eater I do not particularly care for the texture of brown (even brown jasmine). As mentioned above it does have a "nutty" flavor and is "chewier" regardless of how well/properly it is prepared. It changes the flavor and feel of every dish I serve it with. Just a heads up.
Best Answer
Once you get the hang of it, rice is as easy as pasta. One thing you say in the question that may be central to the difficulty you are having is that your lid is "half-closed". For the majority of rice cooking methods, not only should you keep the lid tightly closed, but you shouldn't even open it to check the rice until it has cooked close to long enough that it might be done. Look at the accepted answer here: Rice gets burnt and watery. That answer explains it more thoroughly.