Google-sheets – Median/Percentiles for “histogram-style data”

I couldn't find the trick for the real date cell to work yet, but you could have a work around if nothing else by adding the the single quote (') before the date in a format: YYYY-MM-DD in the Q4 cell.

That single quote means that the it will be parsed as string.

Yes, you can do all of this with Google Apps Script, though it's a pretty large project that arguably should not be someone's first project with GAS. Here is an overview of methods involved, with pointers to documentation.

1. Select a range of cells in a row: getRange.

2. Create a graph based on that range of cells including key and labels: see Embedded Chart creation, Setting chart type, and customizations such as EmbeddedBarChartBuilder.

3. Save image file of that graph in a public location. The only place you can save is Google Drive, as follows:

 var sheet = SpreadsheetApp.getActiveSheet();  // grabs active sheet
 var chart = sheet.getCharts()[0];   // grabs the first chart on the sheet
 var file = DriveApp.createFile(chart.getBlob());   // saves it as a file

You get a PNG file named chart.png; this file can be renamed with setName and made public as follows:

file.setSharing(DriveApp.Access.ANYONE_WITH_LINK, DriveApp.Permission.VIEW);

4. Insert link to image file in sheet where cells are located. Use getUrl to get the URL of the file. Add it to the sheet with setValue. Example:

sheet.getRange(5, 3).setValue(file.getUrl());   // automatically linked

You can make the link look fancy (with custom text) using setFormula('=hyperlink('+file.getUrl()+','+ your_text + ')') instead of setValue.

5. Repeat on next row until blank. Sure, that's what for and while loops are for, among other things such as getDataRange