Will you be so kind and provide screenshots from your ingame statistics and achievements screens?

I have to disagree: in my experience, impossible mission are very often impossible because I chose the wrong mission in the wrong Island with the wrong setup, so it's still my mistake (stategic if not my tactical).
If the possibility of the mission is checked when the mission is created, what would the point of choosing the island / mission / equipement?

Yes. What do you need them for?

Verifying your commitment to this game. Many players questioned balance of Hard difficulty in FTL, yet it's possible to make consecutive wins with every ship with no losses between. Into the Breach may have a similiar approach in terms of hard difficulty balance.

I understand what you're saying, and I sort of agree with you. I'm mostly throwing out ideas, because I think the game is cool, but there's something missing. If the game did check for win possibility, the overall strategy would probably need to be re-evaulauted a bit. But, at the very least, the final battle should be balanced so that any squad without extra items can win. No extra items, but max level + as many reactors as possible would be acceptable. Because at the moment, I'm not convinced that some of the base squads can win the final battle without some random luck. And, as a tactical/strategy game, we should be able to determine whether or not we have mastered the game. Unfortunately, I don't know if I'm good at this game, or just lucky. I've played over 40 hours, and I have like 63% of the achievements. So, maybe once I 100% the game, I will have a better idea. But, for right now, I'm skeptical that this game can be mastered as a skill because there seems to be a fairly strong requirement for luck with certain squads.

You've got some fair points, though I'm totally fine with games in which luck is part of my strategy (for exemple, I like blood bowl, a game in which a bad die roll can end your turn, but it's not dumb luck, you have to prioritize your actions).
Moreover, even a 2-island-playthrough means a choice of 8 extra weapons (excluding pods and perfect islands). If none are helpful, you're being extra-unlucky.
Last but not least - and maybe you haven't unlocked them yet - there are pilots whose special abilities give you extra means of attack.
So yeah, luck is part of the equation, but I still think there are many ways to reduce that part, and, IMO, that is fine.

(for reference, I've unlocked everything,,, but I've still loosing more often than I win)

When you have a fly on water 2 lanes deep across the map and is also next to one of your buildings, there is nothing really you can do without destroying your own building. This is very annoying as I keep coming up against the same situation.

Not sure I understand, because what I do understand seem pretty manageable (unless you play random or custom and have an unbalanced squad).

Like when you destroy the dam. A fly alien can cross the water and threaten a building and there is nothing you can do about it without destroying the building.

.

Like that?

www
wVw
gBg
ggg

w: water
V: Vek
g: ground
B: building

Rift Walkers / Blitzkrieg / Steel Judoka / Flame Behemoths : Ranged should be able to at least move it, just aim next to it.
Rusting Hulks: Brute can smoke it
Blitzkrieg: Prime's weapon can be powered to link building. If it's a 2-HP fly, you can kill it by attacking the building.
Steel Judoka : Science can pull it (or shield the building).
Flame Behemoths: If it's a normal fly, a 1-HP fly or a burning 3-HP one, Prime can push it (or kill it) if you've invested in weapon range (buildings are immune to fire so you won't destroy it).
Frozen Titans: range can freeze it.
Hazardous Mechs: prime can attack and move it or science can push it.

And if none of that is possible... well, loosing one building is (probably) not the end of the world.

(Sorry if I'm underestimating you, I may not have understood after all)

www
wVw
gBg
mm

m=mountain

I have only just started playing so most of that is way over my head, but it does look like I have other options later on in the game. Thanks.

You're welcome.
But even now (I'm assuming you play Rift Walkers), aim the water tile on either side of the Vek with the ranged. It won't do any damage but the vek will be move.