Thinking on cart logic... while it's not clear what we can do yet in a single plane, it seems to me that we've already got enough info to create controllable logic branches vertically.
A "switch statement" (aka case, select, or inspect; simplest version is if-then) would be a vertical pit. For a simple if-then, it's only one level with a single bridge; if the bridge is present (lowered/extended), the mine cart continues across on the same level (Z=0); if the bridge is absent (raised/retracted) the cart drops a level to the tracks below and continues along Z-1. A more complex if-then-else or switch statement could have a fairly arbitrary number of levels, with bridges on each one; the cart falls until hitting a bridge and then continues along on the level that the bridge is present, or hits the "else" clause floor. If you're designing mainly in a plane, you can have a ramp up and an appropriate amount of rollers immediately after the pit-switch to return the cart to the same Z plane and approximate speed.
Logically, this is very slightly less powerful than an arbitrary switch statement in a high level language, as it implies ordered evaluation and can inherently only contain branch statements. If (bridge0) then (tunnel0) elseif (bridge1) then (tunnel1) ... elseif (bridgeN) then (tunnelN) else (tunnelFloor). You can mimic a high level switch statement, however, by having the tunnels be short tracks that each do whatever operation you need done (fill, empty, set on fire, etc.), and then use another pit to collect all the tunnels into a single track again.
Bad attempt at side view:
___ ...___?___ | ____
|...___?___ | _/
|...___?___ | _/
|...___?___ | _/
|______?_________/
Underscore (_) is track in tunnel; ellipsis (...) is bridge which may or may not be there controlled by lever, pressure plate elsewhere, or whatever logic system you want; question mark (?) is the various processes you want to switch between; slash (/) is ramp up with spots for putting acceleration rollers in between (it's not yet clear whether we can have directly powered ramps). The width of the two pits needs to increase as you add depth, as the cart will be heading down in an arc and needs to land on its wheels before it smashes into the far wall; depending on speed the above drawing may need to be stretched horizontally quite a bit.
Edit: Staring at the above, I suspect that the guts of a gravity-powered ripple-carry adder are implied by the concept, although that really isn't my area of expertise.