Default Superscript/Subscript size, by Typeface Designer or InDesign?

The following script will copy the settings for a selected font into the active document's settings. Do note that the information can only be retrieved for TrueType/OpenType fonts! It may be possible for other font types as well (Type 1 fonts, for example, may have this information in their – separate – AFM file) but I do not know how to retrieve that.

The TTF tables actually define more than you can use in InDesign: separate X and Y sizes (the script only uses the Y size), and both X and Y offsets (again, only Y is used). Oh - and you can change the values in the boxes before pressing OK.

Full script is below, a compressed zip is on http://www.jongware.com/binaries/subsupersize.zip

//DESCRIPTION:Find Superscript/subscript sizes
// A Jongware Script 10-Jun-2018
font = app.textDefaults.appliedFont;
if (app.documents.length)
    font = app.activeDocument.textDefaults.appliedFont;
// Any text selection?
if (app.selection.length && app.selection[0].hasOwnProperty("parentStory"))
    font = app.selection[0].parentStory.insertionPoints[0].appliedFont;
if (app.selection.length && app.selection[0].hasOwnProperty("baseline"))
    font = app.selection[0].insertionPoints[0].appliedFont;
// is it an OpenType type?
switch (font.fontType)
{
    case FontTypes.TRUETYPE:
    case FontTypes.OPENTYPE_CFF:
    case FontTypes.OPENTYPE_CID:
    case FontTypes.OPENTYPE_TT:
        subsuper = findSubSuper (font.location);
        break;
    default:
        alert ('Unknown font type; unable to retrieve data for font "'+font.name+'"');
        exit();
}
var aDialog = app.dialogs.add({name:"Super/Sub: "+font.name.replace(/\s+/g,' '), canCancel:true});
with (aDialog)
{
    with(dialogColumns.add())
    {
        staticTexts.add({staticLabel:"Superscript"});
        with (borderPanels.add())
        {
            with (dialogColumns.add())
            {
                with (dialogRows.add())
                {
                    staticTexts.add({staticLabel:"X Size"});
                    staticTexts.add({staticLabel:String(subsuper.SuperscriptXSize)});
                }
                with (dialogRows.add())
                {
                    staticTexts.add({staticLabel:"Y Size"});
                    spysiz = percentEditboxes.add({editValue:subsuper.SuperscriptYSize,minimumValue:-200,maximumValue:200});
                }
                with (dialogRows.add())
                {
                    staticTexts.add({staticLabel:"Y Offset"});
                    spyoff = percentEditboxes.add({editValue:subsuper.SuperscriptYOffset,minimumValue:-200,maximumValue:200});
                }
            }
        }
        staticTexts.add({staticLabel:"Subscript"});
        with (borderPanels.add())
        {
            with (dialogColumns.add())
            {
                with (dialogRows.add())
                {
                    staticTexts.add({staticLabel:"X Size"});
                    staticTexts.add({staticLabel:String(subsuper.SubscriptXSize)});
                }
                with (dialogRows.add())
                {
                    staticTexts.add({staticLabel:"Y Size"});
                    sbysiz = percentEditboxes.add({editValue:subsuper.SubscriptYSize,minimumValue:-200,maximumValue:200});
                }
                with (dialogRows.add())
                {
                    staticTexts.add({staticLabel:"Y Offset"});
                    sbyoff = percentEditboxes.add({editValue:subsuper.SubscriptYOffset,minimumValue:-200,maximumValue:200});
                }
            }
        }
    }
}
if (aDialog.show() == true)
{
    app.activeDocument.textPreferences.properties =
       {superscriptPosition:spyoff.editValue,
        superscriptSize:spysiz.editValue,
        subscriptPosition:sbyoff.editValue,
        subscriptSize:sbysiz.editValue};
}
function findSubSuper (path)
{
    var fontFile, subsuper = {};
    fontFile = File (path);
    if (!fontFile.exists)
    {
        alert ('the file "'+path+'" does not exist');
        exit();
    }
    if (!fontFile.open('r'))
    {
        alert ('the file "'+path+'" could not be opened');
        exit();
    }
    fontFile.encoding = "binary";
    hdr = getUlong(fontFile);
    if (hdr == 0x74746366)    // Truetype collection
    {
        hdr = getUlong(fontFile);
        if (i != 0x00010000 && i != 0x00020000)
        {
            fontFile.close();
            alert ("Not a font type I can use...");
            exit(0);
        }
    } else
    {
        if (hdr != 65536    // Microsoft Truetype
            && hdr != 1330926671)    // Adobe OTF ‘OTTO’
        {
            fontFile.close();
            alert ("Not a font type I can use...");
            exit(0);
        }
    }
    numTable = getUshort (fontFile);
    fontFile.seek (12);
    
    var os2 = null, head = null;
    for (var t=0; t<numTable; t++)
    {
        tableEntry = fontFile.read(16);
        if (tableEntry.substr(0,4) == 'OS/2')
            os2 = tableEntry;
        if (tableEntry.substr(0,4) == 'head')
            head = tableEntry;
    }
    if (!head || !os2)
    {
        fontFile.close();
        alert ("No proper table entries found--yuck");
        exit(0);
    }
    // find head for em units
    offset = (((((head.charCodeAt(8 ) << 8 ) +
        head.charCodeAt(9)) << 8 ) +
        head.charCodeAt(10)) << 8 ) +
        head.charCodeAt(11);
    
    fontFile.seek (offset+9*2);
    unitsPerEm = getUshort (fontFile)/100;
    
    // find OS/2 for settings
    offset = (((((os2.charCodeAt(8 ) << 8 ) +
        os2.charCodeAt(9)) << 8 ) +
        os2.charCodeAt(10)) << 8 ) +
        os2.charCodeAt(11);
    /* the first 5 uint16 entries are not interesting */
    fontFile.seek (offset+5*2);
    subsuper.SubscriptXSize = getUshort (fontFile)/unitsPerEm;
    subsuper.SubscriptYSize = getUshort (fontFile)/unitsPerEm;
    subsuper.SubscriptXOffset = getUshort (fontFile)/unitsPerEm;
    subsuper.SubscriptYOffset = getUshort (fontFile)/unitsPerEm;
    subsuper.SuperscriptXSize = getUshort (fontFile)/unitsPerEm;
    subsuper.SuperscriptYSize = getUshort (fontFile)/unitsPerEm;
    subsuper.SuperscriptXOffset = getUshort (fontFile)/unitsPerEm;
    subsuper.SuperscriptYOffset = getUshort (fontFile)/unitsPerEm;
    fontFile.close();
    return subsuper;
    
    function getUshort (aFile)
    {
        var result = aFile.read(2);
        return (result.charCodeAt(0)<<8) + result.charCodeAt(1);
    }
    
    function getUlong (aFile)
    {
        var result = aFile.read(4);
        return ((((result.charCodeAt(0)<<8 ) + result.charCodeAt(1))<<8 ) + result.charCodeAt(2)<<8 ) + result.charCodeAt(3);
    }
}

I'm not sure that InDesign 3 had support for OpenType, though. Maybe the OP means InDesign CS3?

Technically, InDesign 3 (original CS) was released in 2003.