I am not an Inkscape or Scribus user but for general answers....
Why would the CMYK values in Scribus be so different from the conversion chart values?
"Conversion charts" on websites, are mostly junk, at least for you. The values they display would be based on their profiles and their software, and their algorithms. Each and every computer is going to interpret color based upon its own assigned color profiles in addition to display calibration. I can't give you precise conversion values which are accurate for you if I'm using my digital device to determine the values. All I can do is possibly get "close" to something which may be a visual match for you.
Software color conversions are a very local matter. Each app (or web site) is going to determine what the appropriate RGB/CMYK values may be based upon its own internal conversion algorithms. It's almost impossible to find 2 digital tools which will display identical conversion values since no 2 devices use the same color profile and are calibrated the same. There is no "one value to fit them all".
To be as precise as possible, one should be using a Pantone Matching System Formula Guide - the printed book, not a software tool. Pick the Pantone color you wish, and the Formula Guide (Bridge) will have conversion values to use for RGB and CMYK. Since the printed Formula Guide is not reliant on any color profiles, the values will be more accurate.
(image from WhatTheyThink.com)
Which should I use - the ‘true’ values (if they are indeed true) from the conversion charts, or the Scribus-generated ones which look closer?
This is hard to say. What looks closest to you may not look as close to anyone else on any other computer system. It depends upon the color profiles for each computer. Again digitally there are no "true values". Unless you look at printed samples of a color, you can never be 100% certain the conversion is accurate (This, again, is why the printed Pantone Formula Guides are helpful.)
Is there any issue with having CMYK values which have decimal places, like the Scribus-gen’ed ones? I.e. is it better/more normal to have whole-number values, in basic logo design at least?
Not really any value in the decimals. On press it's very difficult to maintain a 1-2% screen of a color. It takes considerable skill. Therefore trying to maintain a tenth or hundredth of one percent is asking for something nearly impossible. I round values to whole percentages.
Are the CMYK values of 21, 0, 56, 58 more cost-effective, because there’s no magenta being used? I.e. does this reduce the colour count, from a printer’s point of view?
Cost effective? No. 4 color process is 4 color process. It doesn't matter if there is 0% of an ink anywhere, all 4 inks are still on the press and the same number of passes are used. There's no real benefit to simply zeroing out a value for one ink. Although if one of the values is very minute, like 1%, production will thank you for zeroing out that 1% - assuming it doesn't shift color too much. Dropping from something like 5%M to 0M is going to change the final color.
Now with spot color, there can be be value in using 3 spot colors. It can be cheaper to run 3 spot colors as opposed to running 4C-process. It depends on the job and the artwork a great deal. If you have artwork/a logo that can be broken down into 2 or 3 spot colors there's almost always benefit in creating those files, especially if you can do it with only 2 spot colors (based on question... green  and black [just K, not a Pantone black]). But there's never really a reason to go beyond 3 spot colors since the 4C process would kick in, and would almost always be cheaper than 4 spot colors.
Rich black can be a rather ambiguous term.
The term "rich black" indicates artwork which is meant to appear to be just black even though it's not just 100%k. Rich black only applies when more than 1 color of ink will be used.
If a logo is merely 1 color (black)....
I've never in my career constructed a solely "rich black" version for any logo - meaning a logo is only black. Rich black is not one universal value. Each and every print house will have its own CMYK values to use for their "rich black". It is not possible to calculate a "correct" rich black for every print need. Don't waste your time with this for a 1 color logo. Simply create a black version of any logo as 100%k.
If a logo has multiple colors....
If you wanted to deepen any black in your logo artwork then yes, utilizing more inks than just K can help. So for the CMYK version, rather than merely using 100%K for black, using something like 0/40/0/100 will make the black appear to have more of magenta tint to it and deepen it. I personally prefer blues in my blacks. So, I'd tend to go with 40/0/0/100 - or if the logo icon is green.. perhaps 40/0/20/100 - but this is all a matter of preference.
While you can never set the correct "rich black" in the traditional sense of the phrase, you can use something more than just 100%k (keep the total inks below 240% though.)
- For spot color if you are running 371 and K, you can overprint 10% of 371 on top the K areas to deepen the K. While this is not the strict traditional use for the term "rich black", it would still technically be a rich black due to the multiple colors in the K area.
- One has to overprint spot colors. It's not possible to mix spot colors with each other or any CMYK ink.
Do I specify the C in the style guide, or does each printer decide whether to use C or U depending on the paper?
No. The C and U designate stock differences not ink differences. You can't possibly know on what stock the logo will be printed. There are C and U values to try and show the user/reader how the ink will appear on different stock, however the actual ink used doesn't change. Just use the Pantone Number.
Would it be better practice for me to just get the exact corresponding CMYK and RGB values for PMS 371 and use those in the logo (and do it that way for future work)?
Generally speaking, yes.
For me, I find it's often best to find a 3 digit Pantone color I want to use, then, using the printed Bridge Formula Guide pull the RGB and CMYK values based upon that Pantone color. (See first question.) The reason I state "3 digit color" specifically is generally the 3 digit Pantone inks are cheaper than the 4 or 5 digit Pantone inks, as well as possibly being more common/available since the 3 digits are older and have been around longer.
If you're relying on digital conversions to generate values for color, you are simply using "best guess" techniques and can never be certain the values you see will be the same values anyone else sees - since no 2 computer systems have the same monitor calibration or color profiles in use. If you want to trust the values you indicate implicitly, you need printed samples of the values.
I've seen many a style guide with great Pantone values, then clearly the RGB and CMYK values are bad or way off. All this means is whomever created the style guide, used a Pantone color they let their software tell them what the conversion values are. They may have been correct for that user in that instance. But they are wildly off visually on my systems.
If, I use Photoshop and input the designated Pantone color, and then look at the conversion Photoshop states I should use.... I see where they got their values... right there. I've seen a yellow turn clearly into green... a purple turn into clearly a blue, etc. So, I go back to the Pantone value, look it up in a Pantone Bridge Formula Guide, and use the numbers Pantone states I should use - it's generally much, much more accurate.
Having a Pantone Color Bridge Formula Guide on hand is a good tool to add to your stable of tools.
Now, I'm realistic. I know not everyone has or can afford Pantone Formula Guides. If you need to reply on software for the color conversion values, I'd suggest one never use any web site, ever. Use local applications. Find the values the software states are correct. Then try those values on another device when possible - i.e. your desktop states 57C 25M 84Y 40K - so try those values on your phone or a tablet or a different computer and see if you get the same color appearance.