So here is my solution to my problem
DISCLAIMER: this is provided as is, without any warranties or support. Use at your own risk! Back up all data before using!
I made an Excel file with macros that achieve most of what I need through a series of steps. Unfortunately it's not a fully automated solution and some manual operations are still required, but these are warranted in my case as I have over 350 questions in my survey and this saves a lot of time for me. Note that the file only works with a single response. This is not intended for multi-response processing.
The Excel file relies on macros so these must be enabled to be useful. The file can be renamed any way you like. I made the file in Excel 2016, but I suspect it will work in 2013 and 2010, and maybe 2007. It probably won't work in versions prior to 2007, although it might (previous versions don't have the Ribbon interface, where I put the macro buttons).
So here is what I do to achieve a well-formatted export of a response.
Open the Excel file and enable its macros (note that you should always have macros disabled by default). If Excel is not prompting you to enable macros and they don't work, you need to adjust its settings (Google for enabling macros in Excel).
Note: the sheet called Libraries lists question types and the column called Text actually shows which question types the macros will process. The other question types will probably not be processed correctly.
Part 1. Set up the file for a particular survey.
Step 1. In your LimeSurvey, open the "Survey Logic file"
Step 2. Copy the logic file table. I use the Table2Clipboard addon for my Firefox browser for this as it has an option to "Copy whole table"
Step 3. Open the Logic_file sheet in the Excel file and paste the logic file table in cell A1. Excel will show a little icon in the bottom right corner saying "(Ctrl)"; if you click it or press the Ctrl key, it will expand into two options, the last of which is "Match Destination Formatting (M)". This is the one you need to click. This is important because it unmerges any merged cells, removes any images and extra formatting.
Step 4. On the Ribbon toolbar, find the "Custom Tab" tab, click it and there click the third smiley face that says "Generate question list". Excel will ask you if you're sure, press "Yes". Give it a few seconds (minutes) to do its thing. Now you're ready to actually work with responses.
You will need to repeat Part 1 for each new survey or whenever you modify a survey. Once Part 1 is done, you can use it to process multiple responses without repeating the steps above.
Part 2. Work with responses.
Step 1. Go to your LimeSurvey and open (view) the response that you want to process. LimeSurvey will open a table with that particular response. It will have 2 columns:on the left there will be [Question code] + "Question text"; and on the right it will have the answers. You need to copy this table the same way as the logic file table above.
Step 2. Open the RawExport sheet in the Excel file and paste the table into cell B1. Note that it has to be B1, not A1. The first column must remain empty! As with the Logic file table above, select "Match Destination Formatting" in Paste Options. If you're processing multiple responses, just clear the sheet RawExport before pasting a new response (i.e. delete everything on the sheet by selecting all and pressing the Delete key).
Step 3. In the Custom Tab on the Ribbon toolbar, click the "Generate Excel Output report" button (the first smiley face on the left). Give it time to do its thing, it might take a while. The Output sheet will be filled with responses presented in a more or less readable form, with unanswered questions mostly removed and tabular data tabulated. But since this is not very readable, there's one more step remaining, see below.
Step 4. In the same toolbar, click the smiley face in the middle - "Generate HTML report". It will create a file called MyHTML.htm in the same location as your Excel file with exported and formatted results. Before you do it, make sure the attached CSS file (bootstrap.css) is in the same folder as the Excel file, or you will get a "File not found" error. The CSS file is only needed to generate the HTML report. You can actually use any other CSS file you want, as long as you call it bootstrap.css, although I have hard-coded some CSS class attributes in VBA (not a major problem). The CSS will be embedded into the MyHTML.htm file so you won't need the bootstrap.css file to read the HTML file.
This is it
Disclaimer #2: Excel is a registered trademark/product of Microsoft Corporation; bootstrap.css is the work of the nice people from getbootstrap.com.