Learn Turkish: Lütfen şu köpeği muayene eder misiniz?

While ediyor musunuz just asks a factual question ("Are you examining?"), using the aorist tense in a second-person question is the standard Turkish way to make a polite request. It translates nicely to "Would you...?" or "Will you...?".

Why did köpek (dog) change to köpeği?

Two things happen here! First, because the dog is a specific direct object (you are pointing out "that dog"), we need the accusative case suffix -i. Second, adding a vowel suffix to köpek causes the final k to soften into a ğ (soft g), making it köpeği.

Why use şu for "that" instead of o?

While both mean "that", şu is used to draw attention to something the other person might not be looking at yet, or something you are actively pointing at. If you bring a dog into a clinic and point to it, şu is the perfect word to use.

I see etmek is being used in eder misiniz. What does muayene etmek mean exactly?