"Messungen" are usually technical measurements like measuring the temperature or pH value or something like that. "Maße" only refer to length, height and width I think. It's a different context. Hard to explain because both are translated into measurements. I am trying to think of exceptions, because of course there will be exceptions to what I just tried to explain... The only one I can come up with right now is "die Maß" which refers to a litre of beer (for example at Oktoberfest) ;-)
"[die] Messung": singular, doing the measurement of a single item once "[die] Messungen": plural, doing multiple measurements. That can be the measurement of a single item multiple times (temperature every hour as example) or different items of one or more objects (size/weight of a box or a person as examples). "die Maße" (in this lession) are the results (length/size) of multiple measurements (probably length, weidth and height).
Is your dictionary older than 1996? For a start, that's an example of why ss is not interchangeable with ß:
Masse = "mass"
Maße = "measurements"
Additionally, as other comments have already pointed out, a Messung is a "measurement" in the sense of the act of measuring something. A Maß on the other hand is a "measurement" in the sense of the dimension of something that you have discovered by measuring it. Interchanging the two is incorrect.
The first of your suggestions! This questions asks you about YOUR measurements. For your second sentence I'd say something like "Wie lang/breit ist das Objekt/der Gegenstand?" (How long/wide is the object) because I wouldn't say "Wie lang/breit hast du das Objekt gemessen?" (literal translation of your second sentence).
In English, if you walk into a clothing shop and the attendant asks "What are your measurements?" they want to know the physical size of your body: Your height, the circumference of your waist, etc. in centimetres or inches. These measurements could be used to design a custom suit, for example.
If you walk into a clothing shop and the attendant asks "What are your sizes?" they want to know the sizes that are written on the labels of your clothes: Size 8 shoes, 32R trousers, XL shirt, for example.