Does this sound natural in spoken English?

"another plane" instead of "some other plane" Otherwise it sounds very natural to me! :) You might use more airport-specific vocabulary like "as it's heading to the gate" instead of "to the spot where I'm supposed to get off," but that's just a matter of personal preference. It's not like the way you said it screams "non-native speaker."

Sounds fine. Only note is that “the spot where I’m supposed to get off” is a really wordy way of saying “terminal”.

The verb for a plane moving around on the ground is “to taxi”. Also, this should be a single sentence. “Because” is a conjunction that joins two clauses and some teachers will penalize you for starting a sentence that way. “Every time I go on a trip, I can’t fully relax after the plane lands because, as it’s taxiing, I feel like some other plane could come and crash into it.” “Taxiing” is pronounced “taxi-ying”

That reads fine to me. If you said that to me, you'd sound perfectly normal.

I think it's sounds good - but some things are maybe a little strange. When talking about the plane you are in it's more natural to use "us" instead of "it". This emphasises you are worried for your own safety, and not so much about the plane's safety. It's quite common to do this when travelling in vehicles, e.g "They crashed into me" instead of "They crashed into my car (that I was in)". Also I would use "another" instead of "some other", as it's a bit more neutral sounding. So I would say: Every time I go on a trip, I can't fully relax after the plane lands. Because as we're heading to where I'm supposed to get off, I feel like another plane could come and crash into us.

It reads and sounds like normal english to me.

It's a bit clunky, but there's nothing inherently wrong with the two sentences. I think I'd probably say this like "Whenever I fly, I can't relax until the plane's at the gate. I'm always fearful that, as we're taxiing, another plane will hit us."

Personally I would say might instead of could.

This reads fine, but this is not how a native speaker would word this paragraph. In informal language, it would be more like "Every time the plane lands, I can't really relax. I'm afraid the plane will run into another one (or hit another one). There's extra information in the paragraph you provided that, in an in-person context with a native speaker is not needed or necessary. Obviously that's quite informal, but that's the way I would say it-everyone might say it a bit differently.

Yes, except it should be one sentence... "...I can't relax after the plane lands because as it's heading..."

Seeing "spot where I'm supposed to get off" made me smile a little. You could say that it's my personal pet peeve, but I shrug when I notice people that would rather take shortcuts to learn the single best defined word for any sort of message they want to convey. Vocab learning should be a beautiful journey, but trust me when I say you don't wanna know how I got to learn the term "blowjob".

I feel like "taxiing to the gate" is repeated so often by flight crew that it'd come out without much thought. I think the "some other" is absolutely fine, not sure how "another" changes it at all.

“Because as it’s heading to the spot where I’m supposed to get off” is really wordy. You can say “terminal” or “where it stops” instead of “the spot where I’m supposed to get off.”

For me, the clunkiest bit is ‘could come and crash into it’. ‘Come and’ is redundant - ‘I feel like some other plane could crash into it’, because it’s taken as read that the other plane would have to approach (‘come’) first before it could do the crashing bit! Having said that, I’d reword the whole thing as ‘I feel like another plane might crash into it’ - that flows more naturally to my ears. To me ‘come and crash into it’ kinda makes it sound like it would approach on purpose, size up its opportunity kind of thing 😆 and then think ‘aha! Time to crash into that smug taxiing bastard!’

I personally might use some slightly different words, but that's just a matter of idiolect

put a comma after lands and make the b in because small. Starting a sentence with Because is rarely correct.