Short answer to the question in the topic: yes.
It may depend on the airline but e.g. EasyJet explicitly states “including handles and wheels” and I would say it is a general rule.
In your case it may be not allowed. Size may be checked at the gate by having you put your bag into a box/cage of those dimensions. Obviously, if the handle gets in a way, it won’t fit in. But, if it looks like a standard, small carry-on suitcase, that is not bulging, I doubt they will bother with checking.
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