I don't understand why my position in the ranking keeps dropping regardless of what I do.
Just, why.
A good way of looking at it is that you have to maintain position at 6, at least, not 8. Japan and China, once modernized, can easily bump you off the charts if you're not careful. Same thing for the substates if they also modernized. They have a ton of military and, once industrialized, a lot of that as well.
The best thing for a small country is to stack prestige as much as possible. As such, i would think that your research order should be something like this:
Medicine - First thing. Double pop growth is wonderful.
Then continue to research the following until a tech finishes researching in early 1839:
Romanticism
Positivism
Mechanical Production - Gives you access to the Machine Gears (sorry, been a while since i played, this might not be the exact name). These are a major and important source for your production, and having a local production is great.
Clean Coal
Early Railroads
if you finish off in late 1838, for example, go research one of the T1 techs that has had no research (I think Belgium has one or two of those). That way, you won't waste research points that would be lost after one year while saving up the maximum for the next.
Either way, the first thing you do when 1840 hits is research Idealism, then go back and hit the next Aesthetics and Social Thoughts techs just so you can rank up your prestige and educational efficiency. After that, start getting your Industrial techs (Chemistry, Mechanization, Metallurgy, Infrastructure) while throwing in a commerce tech or two Economic Thought also gives prestige bonuses, so if you want to throw that in earlier, you can. Continue to repeat that
In the long game, Medicine and Electricity are the most important techs due to the plus population growth. For colonizing, you need to have nationalism and imperialism, machine guns, and medicines researched by 1870 and ready to go (the only time i would suggest skipping one of the Philosophy policies in Culture would be to get machine guns so that you can be the first to colonize).
As for navy and army techs, these are probably the least important (you should only be fighting wars of choice against uncivilized nations for the first part of the game, anyway). Army can be at least somewhat useful since you might have to fight against the Netherlands (probably will, granted, but definitely at the early game before it matters much), so investing into the Light Armament Tree is never a terrible choice. Another good one might be Army Doctrine just so you can build ports. Also, for the naval side, you must be able to have a level 3 port by 1870, so work in naval tech when you have a lull in research. Also, ironclads are the most efficient colony point boost per naval limit, although with only one port in your homeland, it can be difficult to build up a fleet. Commerce raiders are the #2 choice as of 1870.
If you want, Kersch has an LP on the lparchive.com that is extremely detailed and covers many different things, in far more depth than I've gotten in. It could be useful to you.