can't help + gerund, can't help myself from, can't help but laugh at

Yes, they all mean the same thing. The simplest form is

I can only laugh when he starts singing.

The "help" is a negation term. It implies you have an instinctive behavior and you require "help" to override the instinct. Notice the opposite meanings of

1. I can't help laughing.
2. I can't force laughing.

In the first case, you are laughing. In the second case, you are unable to laugh. That gives some intuition that "help" is the opposite of "force". It is used to mean "deny" or "stop" or "not".

1. I can't help laughing.
2. I can't stop laughing.
3. I can't not be laughing.

Those three mean the same thing.

Then the complexity comes: In your examples 2 and 3, there is a third negation term. "but" and "from" are also negation terms, but they don't do anything. English has a hard time with counting negatives, and putting three negatives together seems to have the same meaning as putting 3 together.

• I can't not laugh = I can help but laugh = I can't help but laugh = I can't help bug not laugh

At some point, it just gets confusing, and the most likely assumed meaning is the one that stops trying to count the negatives. Strictly speaking you can parse those to be opposites of each other. But you are asking for trouble if you expect a native speaker to assume you intended them to do the correct number of negations. The colloquial use of this sort of phrase is "I was laughing" (based on context). Anything else is assumed to be a fancy way of saying that, even if the number of negations technically means "I was not laughing"

The idiomatic constructions can help (X-self from)+ V-ing, and can help but + V are Negative Polarity Items (NPI). That means that, like ever or take long, they can only occur grammatically in a negative environment. Outside a negative context (which includes more than you might imagine), NPIs are ungrammatical.
(Ungrammatical sentences are marked with an asterisk *)

• She hasn't ever been there.

• *She has ever been there

• They said this wouldn't take long.

• *They said this would take long

• I can't help (myself from) laughing when I see it.

• *I can help (myself from) laughing when I see it.

• I can't help but laugh when I see it.

• *I can help but laugh when I see it

And all these idioms have the same meaning (when used grammatically in the negative, at least), so...
Yes, they are all paraphrases of the double negative sentence that you cite.