Are these sentences correct?

They're a little formal in tone--more like an attorney talking to their client than two contemporaries. A few thoughts:

1) Instead of "because most of the offenses---except the murder of your father---are statute-barred," try: "because most of the offenses expire over time. But crimes like the murder of your father never expire."

2) This paragraph is a tough one. It doesn't really ring true for me. Maybe: The rusty old beater pickup didn't even have seatbelts. D. (Did you mean R here?) looked at it in disbelief, especially after he had seen the almost new SUV in the back of the farm.
"After you." D opened the door on the driver's side, gesturing for R. to get in.
"They're watching us," R. hissed. "Now they probably think we're a couple."
"Well, we sort of are." D. retorted, pushing R. forward across the bench seat, the chain rattling against the gear shift as he did.

Good luck with the stories!

I suspect the phrase you're looking for is statute of limitations (as in ". . . most of the offenses--except the murder of your father--are time-barred from prosecution since the statute of limitations has run out.")

It may seem somewhat technical, but it's a bit more accurate and specific.

Groundbelow73 said:

I'm not a native speaker. To improve my language skills I sometimes write stories in English (just for the fun of it.)
Are the following passages correct?

1) "Law became stricter," he explained, "and it became more and more difficult to work with shell companies. Today the group is no longer that powerful, but powerful enough to get us in serious trouble. The problem, however, isn't what they did back then," he pointed out, "because most of the offenses—except the murder of your father—are statute-barred."

Note: My beta reader didn't seem to know the term "time-barred." I rephrased the sentence, but I'm still not sure whether it's correct. In case the term "statute-barred"(or "time-barred") is wrong: you steal something, and years later it's proven. Nevertheless, there won't be filed charges against you because the case is no longer of any interest. Only for murder you can be convicted any time of your life.

2) The truck was an old, dented, rusty pickup that didn't even have seatbelts. D. looked at it in disbelief, especially after he had seen the almost new SUV in the back of the farm.
"After you." He opened the door on the driver's side, gesturing for R. to get in.
"They are watching us," R. hissed. "Now they probably think we are like an old couple that not even separate in order to use the toilet."
"Well, we don’t separate in order to use the toilet," D. retorted, pushing R. forward.
Luckily, it was easy to get in as the pickup was equipped with one long seat in the front instead of two separated seats.

Context: My characters are chained together, which make it impossible for them to get in the car separately.
"On" or "at the driver’s side"? I'm not sure about "like an old couple that not even...," and about the last sentence.
About the old couple--I was thinking about girls, best friends, who go everywhere together, even to the restroom. I don't know whether this is common in the US. In case it is, could you say something like, "Now they probably think we are best buddies who even use the toilet together"?

Thanks in advance.

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1. I'd use 'laws' as if you're talking about one, you'd say 'the law,' or 'that law,' though 'the law' is also collective, as in 'the law of the land,' or 'you've never broken the law?' Isn't English fun?!

I'd also say 'the statute of limitations has run out/expired.' It's common enough to be spoken by pretty much anyone believably. Statute-barred isn't how it's said, and would be confusing, as would time-barred, imo.

2. I don't mind the old couple thing. I'd say bench and bucket seats - but I don't think any old pickups had bucket seats. Even now I think a ton have bench seats, afaik.

I'm fine with the first paragraph, the only bit that isn't fluid is "law became stricter " - would be better as "the law became stricter" or "laws became stricter".

Groundbelow73 said:

2) The truck was an old, dented, rusty pickup that didn't even have seatbelts. D. looked at it in disbelief, especially after he had seen the almost new SUV in the back of the farm.
"After you." He opened the door on the driver's side, gesturing for R. to get in.
"They are watching us," R. hissed. "Now they probably think we are like an old couple that not even separate in order to use the toilet."
"Well, we don’t separate in order to use the toilet," D. retorted, pushing R. forward.
Luckily, it was easy to get in as the pickup was equipped with one long seat in the front instead of two separated seats.

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This one has a few niggles:

-I think you should use more contractions. "They're watching us", "we're like" instead of "they are" and "we are". Speaking without contractions is unusual and only really done for special emphasis. Sometimes it can even sound hostile.

- "D. looked at it in disbelief, especially after he had seen the almost new SUV in the back of the farm." - this sentence structure is a bit off, you're saying he "looked... especially". You can't especially look at something, you just look at it. It's the disbelief that's special, not the looking, so that needs some rewording. That is not really a language mistake, though, plenty of native speakers do things like that!

-the sentence about the old couple is a bit long and slightly awkward. I'd shorten it - but that's more a comment on style than grammar. Where the grammar falls down is "not even separate" - that should be "don't even separate".

That's the only actual error, but if you're up for stylistic comments: I somehow don't like the use of "separate" - can't tell you why, if just feels wrong here. I'd go for something like "we look like some old couple that holds hands to go to the bathroom!" - something simpler and less formal.

Does that help? I have to say, your English is amazing, very idiomatic and fluent. It's a very hard language to pass as a native speaker, but you're really not far off (in your writing at least). A bit more practice and no-one would know

The answers depend on where your story takes place.

Groundbelow73 said:

1) "Law became stricter," he explained, "and it became more and more difficult to work with shell companies. Today the group is no longer that powerful, but powerful enough to get us in serious trouble. The problem, however, isn't what they did back then," he pointed out, "because most of the offenses—except the murder of your father—are statute-barred."

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In the US, the legal term you're looking for is "statute of limitations." As cornflake says, it's widely used and understood by lay people. The equivalent term may differ in the legal systems of other countries.

I would omit the second dialogue tag. It's unnecessary--the reader will remember who's speaking--and distracting, because it makes the reader wonder why it's there.

2) The truck was an old, dented, rusty pickup that didn't even have seatbelts. D. looked at it in disbelief, especially after he had seen the almost new SUV in the back of the farm.

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"After you." He opened the door on the driver's side, gesturing for R. to get in.
"They are watching us," R. hissed. "Now they probably think we are like an old couple that not even separate in order to use the toilet."
"Well, we don’t separate in order to use the toilet," D. retorted, pushing R. forward.

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In the US, one would say "use the bathroom"; "use the toilet" is a more British phrasing.

Culturally, the stereotype in the US (and possibly elsewhere) is for women to use the bathroom in pairs. There's not really such a stereotype for old couples in the US.

I'd strike the "in order" both times in the above dialogue; it's unnecessary and makes the dialogue sound stilted. Also, a more natural way to phrase "an old couple that not even separate" is "an old couple that doesn't even separate."

That's true about they old couples, maybe "a couple of schoolgirls" might work better

Groundbelow73 said:

D.'s answer to it might be, "Well, there's some truth in it, isn't it?"

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That should be "Well, there's some truth in it, isn't there?" or "Well, it's true, isn't it?"

Sorry to niggle! But then I suppose you asked for it (literally).