i don't know exactly what your rectified voltage should be, but using the voltage marking (63V) on C2 as a guide means it's probably about 50V (48V has become an industry standard for industrial controls). so the motor turning slowly at 12V isn't a surprise, but you're right, it turns, aso there's no immediate reason to suspect it has a problem. you probably had a chip work it's way into the controller box and short between the case and the triac. when you put the unit back together, it might be a good idea to stretch a piece of nylon stocking over any openings or vents to keep metal chips from getting inside, but still provide airflow. back to troubleshooting... the diodes look like 1N4004. you can use any diode from the 1N400x series as long as the last digit is equal or higher to that of the original (i.e. 1N4004, 4005, 4006, or 4007) ,so which ever part is most readily available you can use. the differences between them are maximum voltage (PIV) ratings (4001=50V, 4002=100, 4003=200 4004=400, 4005=600, 4006=800, and 4007=1000), which ideally should be twice the highest voltage expected to be encountered in the circuit. so at 50V or so RMS input= 50*1.414=70.7V peak. 70.7Vp *2= 141.4Vp-p. doubling that we get 282V. the next highest step up from this estimate is the 400V rating of the 1N4004. so if you can't find 1N4004s, but can find 1N4007s, they will work. i would check the diodes out of circuit, as well as testing the capacitors out of circuit. lifting one end of the component is sufficient to get it out of circuit. if the triac is shorted, chances are that the diac is also. diacs can take very short current spikes when they hit breakover, but won't handle extended overcurrent. once you get repairs completed, the best test before connecting the motor would be to connect a resistive load where the motor would go. a 25 watt 120V light bulb would be perfect, since you will see the response when you turn the pot. before you start, mark the position that the pot is at with a permanent marker. since the circuit is very similar to a lamp dimmer, you should be able to control the brightness of the bulb with the pot and see it's response.
solder sucker or wick? i use both. the sucker is good for double sided boards with small traces, and the wick (or larger sucker) is generally better for single sided boards with larger traces. both of them take a bit of practice to use properly. for a sucker on this board i would use a large sucker. the larger the "swept" volume of a solder sucker, the better. also, i usually use the soldering iron (preferrably a temperature controlled one) at a higher temperature than i use for soldering the components in place, since the higher temp reduces "time on target" and makes lifting a trace less likely (it sounds counterintuitive, but it works... too low a temperature makes you hold the iron on the pad longer, and is more likely to melt the glue that holds the trace down). also try to avoid touching the iron with the tip of the sucker. the teflon will erode the surface of the tip. i heat the solder with the iron and get it all melted (it helps to add a little bit of solder to improve heat transfer from the tip to the work), align the sucker tip slightly above the work, and in a quick smooth motion, pull the iron away-plop the sucker tip down on the work-and push the release on the sucker all pretty much at the same time. with practice, it moves pretty quickly and smoothly. also make sure the board is held in a stable position, such as with a PC board vise. it's not easy to desolder a board that is in motion.
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