No.

A Barbarian does not benefit from the special damage resistance from the bear totem while in heavy armor.

...but the 6th-level totem choices can benefit a Barbarian wearing Heavy Armor if it doesn't mention that they can't be wearing such armor to gain the benefit.

According to Jeremy Crawford: The intent is that anything that relies on Rage does not benefit a Barbarian wearing Heavy Armor.

Why? I take this ruling to mean a Barbarian is incapable of gaining any and all benefits of Rage while wearing heavy armor, totems included.

To me, this means you simply can't Rage while wearing heavy armor, because there would never be any usefulness or benefit to entering a Heavy Armor Rage; however, Crawford has added more context to his ruling that states that Heavy Armor does not prevent you from entering a rage, only benefiting from it.

As you quoted, it says you gain the benefits of Rage only if not wearing heavy armor... so if you are wearing heavy armor but can not benefit from Rage, you also do not benefit from any features that depend on Rage, such as the 3rd level totem features.

Grappling doesn't care about weight, only size matters.

The grappling rules do not distinguish from grappling a 300lb Medium-sized creature from a 100lb medium creature. If you're grappling a medium-sized princess, it doesn't matter how heavy she is, your speed is always halved, unless the creature is two or more sizes smaller than you.

So, in your example, the Barbarian could only move 25 feet then dash for another 25 feet. And, yes, they can do 5d6 damage, if everything lines up for them. On subsequent rounds, the Barbarian could just move back down, grapple again and dash up 25 feet. Equating to 2d6 damage on subsequent rounds, not that impressive any more.

It should also be noted that carrying/lifting/dragging an object is governed by different rules from carrying (i.e. grappling) a conscious kicking screaming princess.