A lot of late medieval manuals which discuss weapons like the one you're describing simply call it the 'axe' - whether or not it has an axehead. For example the Italian 1409 Fior di battaglia (flower of battle) describes what many modern people would call the pollaxe as simply 'Azza' - Axe. Like wise the Jeu de la hache (game of the axe) from France circa 1400 names the weapon 'Hache' - Axe, despite the drawings clearly showing a hammer on one side and a curved spike on the other with no axe to be found.

Halberd is a German name for a weapon which usually has an axehead on one side and a spike on the other

Polearm is a generic modern term for any weapon which is based around a long pole, it doesnt specify a particular weapon: spears, halberds, pollaxes, and the plain old quarterstaff are all polearms.

As for effectiveness, the Fior di Battaglia specifies that a viable technique with the axe is to just bash someone in the helmet with it, and if that hasnt killed them to just do it again. There isnt really any type of head protection that would let you shrug off a direct blow from a long pole weapon, and still be light enough to allow you to look around, move, or hold your head up properly. The forces involved are too great, and even if your head itself is fine, all that force going through your neck will still put you down. I'm in a club that practices the techniques in these manuals, and we dont do polearm sparring, even with quarterstaves because there's no way to make it safe.

The body has better targets than the parts protected by breastplates or brigandine, I suggest taking a look at the historical manuals that depict how you fight someone in armour to get an idea of what techniques were used.

That'd usually be described as a Lucerne hammer, poleaxe or a bec d'corbin depending on the details of the design such as any other components of the head but i shouldn't get obsessed when it comes with naming polearms. There's a lot more varieties than there are names, there's often grey areas between polearms and its likely medieval people weren't all that picky about what name you used for many designs. It would be somewhat unusual given the nature of medieval armies for large forces to be armed with the exact same weapons, particularly with polearms as there was so greater a variety of weapons of that kind used. Medieval armies after all were not generally raised and equipped centrally by a few large factories as they are today but soldiers would often purchase much of their own equipment from sources locally available to them and so the personal preferences of the individual soldier, what they could afford and what was locally available would go into what exactly their weapons looked like. But an army who's infantry primarily uses polearms of which these varieties of polearm were common would certainly be very plausible if they regularly dealt with armoured enemies. As for the effect fighting well armoured opponents was the primary thing that such polearms were designed for, brigandine and gambeson would offer some protection still but a solid blow would still have a good chance of breaking ribs, skulls ect depending on how much force you managed to put behind the blow, where it lands, how thick the gambeson is and other such factors.