q1.
The capacitance value of a capacitor is pretty meaningless. Large electrolytic capacitors are designed to be +/- 20% around the value written on them. Manufacturers known for their high quality capacitors do make them as close as possible to the value listed on the label, but even so, the actual capacity will vary with several factors when they're actually in use, one of them being the frequency of the circuit they're in.
So don't buy a multimeter just because it advertises being able to measure big capacitors. Buy one that has all the other specs decent.
UT71A looks good on paper but check this forum, i think there's a review and this model didn't do so good.
Uni-T has some models that are great for the price, some are not so great, and it's even possible that two meters in the same series look different inside and it's as if they were made by two different groups of engineers.
For example UT61E is very nice and good for the price, and it's different inside than UT61A.. there's other processors inside and all that.
You can find a review of UT71D and UT61E here :
https://www.youtube.com/playlist?list=PLB46F445E454CD696 (right at the bottom of the playlist)
Anyways, back to question...
Yes, it's possible the meter will be able to measure up to 20 mF but it will probably take a long time for it to actually measure (think 5-10 seconds for such large capacitor).
If you try to measure a bigger capacitor than 20 mF, you can't be sure of what will happen. The manual of the chip inside may say more, or the manual of the meter.. It's possible the meter will measure it but the error may be more than 1.5%, it's possibly the meter will keep trying to measure but never actually show you the value, or it might just say OL or something like that.
But like I said, I don't see why you'd care so much about the capacitance value because it doesn't help you much in real world to know it. It doesn't tell you if a capacitor is faulty in a circuit or not with a good amount of certitude. For that, you'd need an esr meter or some tool that gives a close enough estimation of the ESR of the meter.
Q2.
ESR is really important in circuits that work at high switching frequency. Computer power supplies work at such high frequencies, of anything between 40 kHz to 120 kHz typically, but some power supplies use even higher frequencies.
Generally, the lower the ESR, the better it is, as the voltage coming out the power supply will be smoother and overall it's better. A larger ESR also causes the capacitor to warm inside at a higher degree compared to a capacitor that has lower ESR.
So ideally, you would choose capacitors that have the lowest ESR possible. However, often manufacturers take advantage of the ESR of a capacitor to use that value along with other components to do filtering and other things with the power going through the capacitor. If you change this ESR too much, in rare cases it's possible to make the quality of the power going out worse than before.
For example, if you change the capacitor that had 0.23 Ohm ESR with one that has 0.06, you may see on the oscilloscope that the output looks worse.
So generally, in power supplies, you should replace capacitors with other capacitors that are as close as possible when it comes to ESR and Ripple specs.
3300uF in 10mm diameter are indeed really hard to find, most series that were low esr are now discontinued.. Nichicon HZ and Rubycon MBZ springs to mind.
There are some manufacturers that are willing to produce some series of capacitors in 10mm diameter but you have to order a box of 2-4000 pcs for them to do it.
No, DO NOT solder capacitors with wires. The ESR of the capacitor will change drastically with the wire length, the capacitors have to be soldered as close to the board as possible.
If you have no choice, lower the capacitance value.
But you do have a choice.
2200uF 16v is used for 12v output in your case. Besides 12v, a power supply outputs 5v and 3.3v so the 3300uF 10v capacitor is actually used for 5v.
Why did they use a 10v rated capacitor for 5v when they could have used a 6.3v rated capacitor?
Well, that capacitor was most probably part of that stand-by 5v circuit, which is running 24/7 even when your computer is turned off. So, they probably wanted to fit a capacitor with a larger diameter and a longer lifetime there, and also more heat resistant.
Picture using the computer for a while and then you shut down the computer - the power supply will be hot and now all the fans inside the computer moving some air are turned off - while the power supply cools down, the capacitors will be subjected to heat.
All capacitors have a lifetime rating... an average capacitor has let's say 2000 hours @ 105c . That means the capacitor is guaranteed to be within the specs in the datasheet for 2000 hours if it's in an environment where the temperature is 105c.
The temp. is not that high in computers though. So manufacturers have a sort of rule, which says for each 10C degrees less than the 105c rating, the lifetime rating can be doubled. So a 2000h @ 105c is 4000h @ 95c , 8000h @ 85c , 16000h @ 75c , 32000h @ 65c and so on.
Capacitors with larger diameter can also tolerate heat better, so it's quite possible a 3300uF 6.3v capacitor with 8mm diameter had only 1000h @ 105c rating, while a 3300uF 10v with 10mm diameter was rated for 2000h @ 105c
Inside a power supply with no fans, it's quite normal for a capacitor to stay at 40-60c all the time, so the manufacturer probably looked at the capacitors they could use and decided 3300uF 6.3v had only 1000h @ 105c and that was too little life, so they went with the 10v rated capacitor.
There's nothing stopping you from using the 6.3v rated capacitor instead, as long as you're aware it may not last those 30000 hours but only 20-25000 hours of usages.
For example, you should still be able to buy Nichicon HM series 3300uF 6.3v (Farnell/Newark still has them) even though it's also a discontinued series. They're 2000h @ 15c if I remember correctly.
If you want, you can go instead with a 2700uF 6.3v Panasonic FR capacitor which also has long life, 10k hours... it's a bit too low esr at 0.018 if you say the old one was 0.23 ohm, but imho it's worth the risk :
http://ro.farnell.com/panasonic/eeufr0j272l/capacitor-radial-6-3v-2700uf/dp/1800611And 2700uF is close enough to 3300uF that I don't think the quality of the power your psu provides will be affected much.
You can go lower, instead of 3300uF, but you shouldn't go lower than 2200uF.
later edit: As for ripple current, that tells you how much current ripple the capacitor can handle. So unless you know how much current goes through the capacitor, the safest thing would be to replace existing capacitor with one that has a ripple current value very close to the old one's value (let's say +/- 5-10%) or much higher value.
More is better when it comes to current ripple, less is better when it comes to ESR.
q3
There's lots of software.. Spice , LTspice comes to mind