I understand that there's a lack of consensus among the Greek grammarians about this, but it would be helpful for us language learners if the Duolingo Greek team would choose one standard or the other, and apply it consistently here. In a highly-inflected language like Greek, when the presence or absence of a single phoneme can mean a real difference in case, gender, number or tense, an apparently random variation like this one can be maddening to the foreign language learner.
Personally, I would be happy if everyone adopted the "masculine articles never drop the ν" rule - it's much easier for me to learn! But I'm also an academic linguist, not a prescriptivist one. The "correct" form of the language, to our way of thinking, is not the form that the prescriptivists dictate but the language that native speakers actually use. If (most) native Greek speakers actually do drop the ν from τον and έναν before a certain set of consonants, I can learn that too - if that rule was applied consistently here.
That would be great, because this is currently very confusing. I just finished a printed card with the very same phrase that used "τον σκύλο" and then was marked wrong on this "type what you hear" one for using it. To be fair, I cannot hear the -ν sound. But these "type what you hear" cards never actually want you to type what you hear. They want you to type what you are supposed to hear.
There is no difference. There are some verbs in Greek called "συνηρημένα ρήματα" that have two ways of telling them. Verbs that end in -αω (such as αγαπάω) fall in this category. Εγώ αγαπάω/αγαπώ, εσυ αγαπάς, αυτός αγαπάει/αγαπά, εμείς αγαπάμε/αγαπούμε, εσείς αγαπάτε, αυτοί αγαπάν(ε)/αγαπούν(ε)