Part 2, please read part 1 first

"I've" is "I have", which is the present tense. English does not use the present tense to describe the past as much as Italian does, though it does do it occasionally. (Particularly when you're telling a story of an incident like this; it helps to bring the listener into the action more.) However the problem here is that you have already used the past tense; "I didn't know why". It's better to stay with either present or past; when you mix them like that it can sound disjointed. I therefore changed "I've" to the past form of "I had".

The security guard asked me to take off my shoes, and that's what I did, but the alarm rang again. Everyone who stood there have started to look at me with suspicion.

1/ This would take a lot more space than I have to explain, but the way you had this you would need "who" in this case. At the risk of boring you it's called a "relative pronoun". You have already said "everybody", but that raises the question of who that actually is. The answer to that is "everybody who stood there". A relative pronoun gives more information about a person or object that you have mentioned, in this case "everybody".

There are however other ways that you can express this, such as "Everyone stood there looking at me with suspicion", but that's a very long discussion.

2/ "Have" is a present tense verb; in this case you would not be using it to describe a completely past event. You would use it if the sentence had some connection with the present; for example "I have started to write a novel". You started in the past, but are still doing it now.

3/ You don't "look" someone but rather "look at" them. There are however other verbs like "watch", which don't need (and can't use) "at" after them; "everyone started to watch me with suspicion".

The guards were looking in my hand baggage, and one of them opened the laptop case and found a bunch of money. I put them it there because I've never brought a wallet with me.

1/ Money is a collective noun, so it's singular rather than plural. ("It" rather than "them".) "Cash" would be the same. However "banknotes" or "Euros" would be plural.

2/ I'm not sure about "I've never". If you have never taken a wallet with you on any trip, then "I've never" is OK. If you mean just on this trip because it's by plane, it would be "I had not".

3/ I recommend ending the sentence at that point.

I tried to explain, but I think they took me for some kind of drug dealer. After a while, and a lot of questions, they left me go. Eventually, I was getting myself back in order, when I put a hand in my pocket and felt the cold of the coins... I've I had forgotten to put the change away when I bought a cup of coffee... Really? Everything for some change that they never even found? It was so embarrassing...

1/ "I've" vs "I had" is the same issue that I mentioned previously.

2/ "That they even found" doesn't work here; that would be for a case where they had found something. In this case "never even" is the right expression.

Part 1 of 2

Airport Story

1/ I suggest trying to keep the number of commas in a sentence to no more than two, unless there is a list of items in it. Otherwise your readers are constantly stopping and starting, which makes reading difficult.

2/ If you say "that time" you usually have to have given some indication of what that time was; for example "Last July I travelled by plane. Normally I travel by train but that time I got a good discount... etc" When you are not specifying when it happened it's better to say "one time".

3/ You can say "I had occasion to do something". It's a little old fashioned and formal, but it's perfectly OK. However it would not be used with "the". "The occasion" is something that happened at a specific time. "Had occasion" is similar to "Had reason to", which is probably a more modern and common way of saying it. "Had the opportunity" is probably even better.

4/ La bella Frecciarossa rocks. There is no earthly way I would catch a plane over one of those. But that's an issue of choice, not language.

So, I was at the Catania airport, waiting in a queue for the security controls.

Ah, Catania... yes, maybe la mia Freccia wasn't an option. 1/ It's not completely wrong but it's more common to just say "Catania airport" rather than "the Catania airport". You would normally use "the" if you were distinguishing it from a number of other airports that you might have been at. 2/ You wait in a queue. You can wait "in line", which is probably where you got "in queue" from, though you should probably regard "in line" as an idiom. You can also wait in a line; when you say it that way both "line" and "queue" are serving as nouns, and need an article ("a" or "the") to accompany them.

When it was my turn, I passed the check and the alarm starting started to ring.

1/ You can't say "when was" because you don't have a noun or pronoun to say what "was". In this case it is usual to say "when it was my turn"

2/ "Starting" is the gerundio form (like "suonando"); in this case you need the passato form (like suonato) since you're talking about an event in the past.