Well i never calculate torpedo solutions by hand, i usualy plot things out using the bearing, AoB and distance tools built into the game (i dont use the speed tool as its typicaly woefully inacurate), and I calculate speed using the
3 minute rule as you do. But i punch that data in to the TDC (i also make sure i lock the target up in the periscope/uzo so it will track the ship in the tdc) make sure everything looks right and the target is being correctly tracked. Im not sure why your trying to manualy calculate it using algebra, as thats what the TDC is designed to do for you (and did for real life captains as i understand it). All it needs is range, speed, angle of bow, and the sighting mechanism locked onto the target (or held steady by you on it)
As for shooting position i typicaly get in close to the target, usualy facing directly across its path (90 degree angle) about 300m away from it for impact torps, for magnetics ill usualy face more inwards towards the target to give the torpedo more time under the target's keel (i face towards rather then away because it gives me the chance of another accurate shot).
I set the fish if im using impact about 1-4m depth depending on the ship's draft (1m if the weather is very bad), if im using magnetic ill set the depth 0.5m to 2m below the the listed draft of the ship (listed in the id book in game), 0.5m if the wind speed and wave action is low to nil, 2m in heavy storms (though i usualy only use 1m depth impact in bad weather as the target ship and torpedo shift depth to much due to waves for magnetics to be reliable)
As for taking the shot i manualy open the torpedo doors (always do this or it will delay the shot by about 5+ seconds and make it arrive 5+ seconds late) and fire when the torpedo bearing reaches 000 for a fore tube shot, or 180 for aft tube shot (making the torpedo turn delays its arival and throws off the shot).
Lastly as for what to target, well ive found the engine room works best, followed second by the fuel bunker (for military ships engine room, fuel bunker, ammunition magazines). Some mods tell you where the spots are on each ship in the ID manual, if your running vanilla, generaly the engine room is right under the ship's funnel, and the fuel bunker right infront of it towards the bow. If the ship has multiple funnels i would aim in the middle of them. If your using Magnetics i would typicaly aim for the dead center of the ship (as thats usualy the best place to break its keel or spine and cause it to break in half).
If your using the TDC method, you can also adjust your aim just by moving the scope crosshairs to the point you want to hit and keeping it there when you fire, or do the opposite and line up your scope so the torpedo bearing reads 000, then wait for the ship to cross your scope and fire when the right part of it crosses the centerline of the scope. With practice you can hit the exact point you were aiming at almost every single time.
If ya like
here is a picture of a real WW2 Uboat's TDC with the cover taken off to show the inner workings, the dials look kind of familiar dont they
Oh another suggesting if your learning to shoot is in the realism settings leave the external view on, and also the option that pops up a window showing the torpedo when its about to hit (action cam?). This way you know exactly what your torpedo did, and what went right/wrong and why.