While it seems Fantasy Flight Games has never explicitly said that Eldritch Horror expansions can be combined, the consensus of players is that it is perfectly fine to play Eldritch Horror with multiple expansions.
No expansion is incompatible with another expansion
Thus far, no expansion directly conflicts with another expansion. For instance, there is no expansion that introduces mechanic X, and another expansion that introduces mechanic Y such that mechanics X and Y cannot be played at the same time.
Most expansion cards can be shuffled in without issue
Because new cards are either Ancient One specific or shuffle naturally into existing decks (e.g. Common Item), there is no conflict between them. While there is some deck dilution, it doesn't seem to be a problem yet.
Multiple expansion packs on the other hand mesh together very naturally and easily. There's no reason to separate base game and expansion components - just shuffle them together. The expansion-specific mechanisms, new spell types, tasks and unique assets and impairments, enter the game through cards in the common decks.
Since you use a random subset of mythos cards, each enemy had their own set of special cards, and the map boards only come out with one enemy and one prelude card, the bloat isn't as bad.
However, each expansion does add more conditions, more spells, more enemy counters, more assets, and so on and those do suffer some bloat. As already said, this isn't excessive - but it could be problematic eventually
Investigators usually lack a reason to go to multiple sideboards
It does seem like having multiple sideboards tends to lead to a situation where nobody wants to go to them. From Reddit:
I own all the current expansions and always play with all of them. Sideboards rarely come out since they are tied to a specific Ancient One or Prelude card. If you do use a side board its usually one and VERY rarely two.
The thing is that there's not really much reason to go to the side boards if you don't have a mission there. Sure, you can throw them down and make them available, but rarely will anybody take the detour because you don't have anything to do there (though the Antarctic might be handy as a shortcut?).
It's really the prelude cards and ancient ones that give you stuff to do on those side boards that make it make sense to have them in the first place.
Cards that are only relevant to a specific sideboard should never come up if you leave them in
You mentioned what to do about cards that are only relevant to a certain sideboard, such as the Hypothermia condition and the Sled Dog asset, which only work on the Mountains of Madness sideboard. Fortunately, a BoardGameGeek user described how this is a non-issue:
Cards with the same backs can be mixed together without issues.
Double sided cards like spells, conditions as well as unique assets will all have their own decks and can also be mixed together.
Don't worry about drawing a Dog Sled if you aren't playing with the Antarctica side board, as finding a unique asset you are usually directed to draw a certain type of card... which is never ally.
The Antarctic research cards are unique backed, thus they will create their own deck, and will only be needed when using the Antarctica sideboard.
Should you somehow wind up with gaining a useless card, another user described a pretty simple way to resolve it:
All of my games (quite a few now already) have been played with Base, FL, MoM, and SR, and yet this situation has yet to arise for me. However, my gaming group has discussed it, and we all agreed if it ever came up, we'd just discard it as irrelevant and draw again.