Yes, you can have different totem animals for different level abilities.
Though few people discuss multi-totemed Barbarians, this is the RAW of the class's wording. Consider the 10th level ability, Spirit Walker:
At 10th level, you can cast the commune with nature spell, but only as
a ritual. When you do so, a spiritual version of one of the animals
you chose for Totem Spirit or Aspect of the Beast appears to you to
convey the information you seek.
(emphasis mine)
This quite unambiguously conveys the idea that you can make different totem choices at different levels, and that they do not have to agree.
RAW, no.
Or, answering the body question instead, the spell ignores these features.
Spells do (only) what they say. The spell does not force a saving throw. For the monster's feature to be a "Specific beats general" case, it should explicitly state something along the lines of:
If this creature is affected by the sleep spell, it gets a X saving throw. The consequences for failing are Y. The consequences for succeeding are Z.
Neither of them do, and, as far as I'm aware, no creature or class has a feature worded like that, i.e., no feature states that "they get a saving throw" in any way. They simply have advantage against saving throws that already exist, which sleep doesn't provide.
A similar case is
The mirror image spell has no effect on magic missile, which doesn't involve an attack.
So, we could answer with the same logic:
The bugbear chief's feature Heart of Hruggek has no effect on sleep, which doesn't involve a saving throw.
Additional conjecture for my answer is that you would have to adjudicate what type of saving throw? Is it Wis? Con? What happens if the creature succeeds the saving throw? Is it simply not affected? Does the HP from sleep's HP pool that would have been used for that monster get reduced anyway? While these answers might be easy to adjudicate, none of them is answered by neither the spell description or the feature. Spells are (supposed to be) clear (that's the whole point of "Spells do what they say"), so, if you have to adjudicate so many things, this is probably not how the effect should be resolved.
As for "Why does the Bugbear Chief have a feature like that then?", I've created this question. NautArch has already provided examples. One of them, the symbol spell, has a possible effect that states:
Sleep. Each target must make a Wisdom saving throw and falls unconscious for 10 minutes on a failed save. A creature awakens if it takes damage or if someone uses an action to shake or slap it awake.
Best Answer
Change the Description of a Normal Attack
If a player just wants to an damage opponent in a cool way, it's easiest to conduct it as the PC's normal attack, but describe it however you want (without any mechanical change).
Alternately, if the player wants to move an opponent to another square, you can use the Shove or Moving a Grappled Creature options (re-flavoring as needed).
Story time: last night my barbarian player wanted to clobber the Big-Bad, using a nearby minion as a club (as you do). I simply said "Sure. You have two attacks, so we'll assign one to each". He hit with both (crit'd actually) and we described the glorious carnage exactly as he specified.
If he had only one attack, we probably would've still described it mostly the same (but only one opponent would've lost HP). Indeed, players really don't even need permission to describe their attacks in a cool way, since they are just following the normal rules. Plus, if the DM likes it: he is free to give advantage to the attack(s).