[RPG] cantrip or spell that grants resistance to damage from natural attacks in the way Blade Ward does for weapon damage

The reference you're looking for is on page 197 of the Player's Handbook, under, naturally enough, Damage Resistance and Vulnerability. Specifically the third paragraph states:

Multiple instances of resistance or vulnerability that affect the same damage type count as only one instance.

So the Bear Totem Barbarian still takes half damage from bludgeoning, slashing, or piercing attacks.

Yes.

On your turn, you can use your action to cast a cantrip that you know. There's no limitation on how many times or how often you can use your cantrips. Your movement will be impacted only in that you will not be able to use your action to dash (but, as you identify, you can still dash with your bonus action).

That covers what you asked about in the question, but here are some other concerns that might arise as a result of casting a cantrip every round:

1. You can't use your action for anything else. You may want to use your action to make a perception check, to dash, disengage or hide on top of what Cunning Action allows, or to interact with more than one object in a round. You won't be able to do any of these things in a round where you cast Blade ward.
2. You need a free hand for the somatic component. Unless you have the War Caster feat, you need one hand free to perform the somatic components of the spell. So you could not hold two weapons, or a weapon and shield, or a weapon and a wand of fly or whatever while casting this spell. A two-handed weapon is okay though- you can hold it in one hand while casting.
3. You need to speak the verbal component. You may alert enemies to your presence when you speak the verbal component of the spell. There is basically no guidance in the game about how loudly you need to pronounce the spell, so talk to your GM if you are concerned about this.
4. You might scare people. Casting Blade Ward every six seconds suggests that you expect that you might be attacked in the that period of time. This is probably fine while you are doing aerial recon, but potentially scary and off-putting if you are doing it while shopping in town. Also note that many NPCs may not be able to identify what spell you are casting. From their point of view, it could just as easily be Dominate Person or Fireball.
5. You may annoy your fellow party members. I'm just imagining spending all day next you a guy going "blade ward... blade ward... blade ward..." ALL DAY LONG. It sounds obnoxious, but this is a roleplaying opportunity more than a reason not to do it. It definitely says something about your character if you are the type of person that would want to do this.
6. You might be distracted. I don't think that there is any RAW support for this idea, but anecdotally some GMs might decide that you are focusing too much of your attention on casting the spell, and give you a penalty to perception or social skills as a result. Again, talk to your GM.