A couple of things to note. As someone else said, do not use the bismuth solders. I think the circuit will probably operate just fine if you use them, as they are conductive, but they are very brittle and will lead to problems in the near future. As to how easy they are to solder with, I don't know, and don't plan to find out. I personally have only used the lead-based solder, in both pace form and in wire. Lead is indeed a severe health problem, especially for children whose brains are still developing. A bit less so for adults, but still not a good thing to be exposed to. But if you are doing things correctly, you should not be getting the paste on your hands. Regardless, don't do soldering where you handle food, and always wash your hands when you are done soldering. With those precautions you should not have any health concerns.
As far as soldering irons go, there is a correct temperature, and going above it is not necessarily a good thing. But you don't need a high dollar iron, From what I have read even the Chinese temperature controlled irons are not that bad, and are available quite cheaply. A Hakko is probably even better, and still affordable by most. Personally I really like my Metcal soldering and desoldering equipment, but that is a bit expensive and less you're a really serious hobbyist or have a lot of money.
As far as the SW and BST terminals, you need to understand what those are for. The SW is the output from the internal FETs, it sends the pulses of power out towards the inductor when one FETs is turned on, and acts as a diode when the FET is turned off. It needs to go to the inductor and to one side of the boost capacitor, and nowhere else. The BST terminal gets connected to the SW terminal by 8.1 µF capacitor, that set up works as a charge pump to provide a higher gate drive voltage for the output FET. It should only be connected to SW through a capacitor, not directly.
As far as capacitors go, for input and output I used 1206 (inch) sized components of the same values you chose. The spec sheet for the regulator specifies those values to be acceptable in most cases, it certainly seems it is specifying the rated value of the capacitors, not their effective capacitance operating voltage. That is how I interpret it, and it worked fine in both of my applications. I also use an inductor which was about the same physical size as the one you used, I can give you the part number if you like.
Since you've got the entire circuit bill anyway, at least that's what I understand, there is something else I suggest you do. Before fixing the problem with the regulator, make sure the rest of your circuit works exactly the way you want. Supply with power from an external supply, program it, hook it up, and see how it does. If it does not work correctly, you can also try to repair those defects on your neck circuit board. It would be a shame to get the regulator working and still need to order another board.
One other thing I should probably mention. You say this is going to be a filament driver, I suppose that could be a lightbulb, but I am assuming it is actually a vacuum tube? If so, is this the final output tube, or a low-level amplifier? I am a little concerned that switching noise on a filament may show up on the output. Also, is the tube designed for use with DC on the filament?