2S is 9 Volts then? Is there a chance of the motors burning out? 2S is two cells in series. For most lithium chemistries (LiFePO4 excluded) each cell is a nominal 3.7V. 2S is therefore 7.4V nominal (when fully charged it'll go up to around 8.4V, i.e. 4.2V per cell, but it won't stay there long). The motors won't "burn out" as such but, as is a fact of life with brushed motors, the brushes will eventually wear away. The stock motors have cheap copper brushes which wear away faster than higher quality carbon or silver brushes in other motors (although you can swap-in a standard carbon brush pack). 3S will wear them out faster than 2S which will wear them out faster than 6V. Most people run them on 3S for long periods of time without any issue.
How many do I need and are they all the same type? In the guide, the person use 2 different types, so I'm confused about that..
You need three microswitches, one for the acceleration trigger, one for the main trigger and one for the pusher cycle control. Form factor isn't critical as long as they are rated adequately. If you're only running the stock motors at 2S (or at a pinch 3S) you can get away with using the stock switches (if you bridge the contacts as shown in that guide) but they won't have as nice a trigger feel. Notice in that guide toruk still used an uprated switch on the acceleration trigger as that has to carry twice the current of the other switches (since it's controlling two motors instead of just one).
The second one is related to diodes.
The diodes can be removed from the stock PCB you don't need them. What they do is reduce the flywheel speed when not firing either to reduce noise or to prevent dart from having a higher muzzle velocity than the following darts (diodes have a forward voltage drop of around 0.7V). Diodes can be used to reduce pusher cycle speed if you find that it overruns it's stop point but I'd recommend higher current diodes for that purpose.
Oh, and another question, did you write the blog?
No, that would be /u/torukmakto4. Note that the soldering iron rating is a "recommended" rating. You could do the job with a 15W iron but it's unlikely that you'd get very good solder joins as it won't have enough power to maintain the tip temperature. It also depends on the quality of the iron. A good 30W iron is better than a good 25W iron but a good 25W iron is much better than a cheap 30W iron (partly because a lot of cheap irons don't live up to their rating and/or have poor quality tips/components). A conical tip is just that, a tip that has a conical shape as opposed to a "spade tip". Don't get too hung up on that when buying an iron as any decent iron will let you replace the tip anyway.
Going back to previous comments I also wouldn't recommend Tamiya motors in a Rapidstrike. Not only do they have a low torque figure (less than what you'd get from some Nerf motors at stock voltage) but they also demand pretty high currents. High current = more losses due to resistance in the wires and mean that you need higher discharge packs. As /u/foam_data pointed out low torque is bad for rate of fire and cycle control aswell but also for motor spin-up time.