Thanks for the video! I will watch in awe at my lack of skills and equipment.
You don't need most of it. I have soldered 0402 size SMD passives even with a bulky, mains powered unregulated iron with a fine tip mounted on it. Not saying that there is something wrong with thermal regulation or a smaller iron wouldn't be better

Short length makes it easier to manipulate the tip with precision.
You need something to hold the component in place, I prefer a tiny flat screwdriver but some people use tweezers. Clean the pads of leftover solder (flux helps), put the component in place, hold it down, solder one end, solder the other end. If something goes wrong, put the iron to the side and heat both ends (add more solder if necessary), when it starts to move freely by itself and not before, wipe it off with the iron. Practice everything on a sacrificial PCB

Beware that those things are tiny and just love to fall on the floor.
As for size of those elements, I guess the small ones are 0402 and the big capacitor was 0805. Find definitions and measure similar items on the board if in doubt. Other candidates are 0603 and 1206.
I'll investigate the area you mentioned and purchase a cheap LCR.
I usually don't have the luxury of buying a second PCB to measure every component so in my experience a more useful gear is a DMM with continuity test to trace the circuit and figure out how it works what should go where. You will probably need it soon if you want to do repairs.
No-O-0... !
I would strongly suggest that being now 'out of the woods'
don't do any LCR readings just yet, just in case mobo #2 is particular about being prodded
and you may have TWO books to close 
There is actually a bigger problem: those capacitors would certainly need to be removed from the circuit to measure them, there is likely many more caps in parallel with them and a lot of other things so in-circuit measurement would be completely bogus.
Resistors
may need to be removed too. For starters, I always measure resistors in both directions when in circuit. Even if the results agree, they may still be incorrect sometimes.
Also, if you desolder something from a working board, better don't lose it. And if you try to measure a 0402 component by squeezing it between two probes, it's very easy to send it flying far away.