Is it bad to skip gears when downshifting manually, e.g. from 4th directly down to 2nd etc?
If your car is slowing down, and you are at a "safe" speed for 2nd gear, then its perfectly normal to skip gears. If you're going the wrong speed and the engine RPMs aren't even close to what they need to be, the syncros might have a hard time lining everything up and you could hear a little grinding, and cause a little wear, but it still shouldn't cause any major damage if you only do it accidentally a few times.
This is most noticeable in many cars when trying to shift into 1st while the car is still moving because it's expected that you won't need 1st unless you come to a stop. This matches up with what I mentioned about matching speed/RPM.
Gearbox impact of skipping gears driving downwards
Semi-related to the second paragraph: Is this why some cars don't let you shift into first gear while the car is moving? My 2000 Renault Laguna prevented that, but the 2006 Volvo V70 is happy to let me (of course then you have to make sure to not release the clutch pedal at the wrong moment, or I'm sure there will be wear on various moving parts). I found it incredibly annoying in the Renault as I couldn't "pre-shift" into first gear approaching a red light or stop sign even if I knew I wanted first gear, but had to wait until the car had stopped completely.– userOct 20, 2016 at 13:06
@MichaelKjörling, yes, the transmission has something called "synchros" that allow you to easily change gears while everything is in motion. Like this question shows, some cars make it very hard to shift into first while moving. The ability to easily shift into first while in motion would add parts and complexity, so that ability is not planned for. I guess it's possible that a transmission would make it impossible, but I'm not sure.– JPhi1618Oct 20, 2016 at 14:32
Except that in my case, the clutch pedal was fully in, so intuitively, there shouldn't have been more things moving when switching from 2nd to 1st gear than when switching from 1st to 2nd gear. I don't think I ever tried to actually force it, but it was at least distinctly more difficult shifting into 1st gear than into any other gear when the car was not at a full stop. Maybe I'll ask a full question on this later.– userOct 20, 2016 at 14:42
@MichaelKjörling, sounds like a good question.– JPhi1618Oct 20, 2016 at 15:33
1st is sometimes not even synchro, and even if it is, it is hard to "spin up" to the necessary rotational speed.– mckenzmOct 21, 2016 at 3:34
It's a recognized advanced driving technique and is called "Block Shifting" or "Block Changing". Provided you are proficient in matching RPM and don't rush the 'box, it's not only fine but there are times when it's preferable.
Some more information from the Bristol arm of the Institute of Advanced Motorists
In the long run, skipping gears when downshifting WILL cause more wear on the synchronizers if you don't double clutch, especially if you are skipping to a low gear for your speed like skipping from 5th to 2nd at 50 MPH. If you do this often as a normal part of driving, your synchronizers likely won't last as long as you would like. Skipping down gears at lower speeds and RPMs isn't as much of a concern, but you will have to be the judge of what is normal use vs abuse for your vehicle and driving conditions. If in doubt, I'd err on the side of not skipping gears when downshifting or double clutching.
One way you can get an idea of how much stress you are putting on the synchronizers is to pay close attention to what the transmission feels like when you shift it. Remember that the longer a shift takes and the harder you have to push on the shifter to get it into gear, the more wear you are causing. So if the transmission goes into gear quickly and smoothly with little force, you likely aren't causing much wear. But if it resists and feels like it's trying to block you out, that's a good indication that you are likely stressing the transmission and causing unnecessary wear.
My car's owners manual says to not skip gears and I know of at least one manufacturer who created a service bulletin specifically about customers ruining their synchronizers by skip shifting, so there is definitely a risk of causing excessive wear when skipping gears if done poorly or taken to an extreme.