My "theory" isn't brilliant or even useful, just that there's a glitch in the conversion and ID thinks all the text has a stroke, for some reason, so every time we backspace into the *old* formatting, ID applies a stroke to the whole paragraph.
It's been a looong time since I have worked in an INDD file that started its life in QXP, but I am totally convinced that your theory is correct. It is, in fact, an artifact of the conversion process, where Q2ID tries to simulate something in Quark by adding stroke in InDesign. (I don't recall if InDesign's native ability to import old Quark docs caused the same conversion error.)
So, to "get it to stop somehow" is going to require more than one step. If you have lots of files to process (and it sounds like you do) then it would most likely be worth it to wrap these steps up into a single script, and to run that script each time you open up a converted file.
1) Turn off all strokes on all text throughout the document.
2) Turn off all strokes in all paragraph and character styles
These are basically the steps outlined by Rene below. I have written this little script to turn off all strokes in all paragraph and character styles, as it has been my project this last year to Get Fast At Javascript, and The Jedi has already posted in this thread. So: here's my little script to turn all stroke weights to zero in all paragraph and character styles.
var myDoc = app.activeDocument;
ParaStyle_Remove_Stroke();
CharStyle_Remove_Stroke();
function ParaStyle_Remove_Stroke(){
var countAllParagraphStyles = myDoc.allParagraphStyles.length;
for(var a=2;a<myDoc.allParagraphStyles.length;a++){
try{
myDoc.allParagraphStyles[a].strokeWeight = 0;
}catch(e){}
}
}
function CharStyle_Remove_Stroke(){
var countAllCharacterStyles = myDoc.allCharacterStyles.length;
for(var a=1;a<myDoc.allCharacterStyles.length;a++){
try{
myDoc.allCharacterStyles[a].strokeWeight = 0;
}catch(e){}
}
}