Calling as in on the phone, not in class, right? If so, confusing English translation, given that the latter would actually make more sense in real life. Ideally, the default verb used to translate מתקשר here would be changed to something less ambiguous.
Not exactly. להתקשר means 'to telephone'. 'To contact' would probably be ליצור קשר (literally: "to create contact").
How about connecting with someone through email or text message? Can you say:
"הוא מתקשר עליו דרך אמייל"
"הוא מתקשר עליו בטקסט"?
If not, how could you say those correctly?
Because it's the female verb form.
Present tense / participle, feminine, singular:
It means “to.” Since it is a preposition, when you use it, you don't need את anymore.