Through generally indicates transit from one end of something to another (often, but not always, the opposite end). This can involve literal passage through space or time ("We drove through Texas", "I waited through the night"), or figurative movement through a system or process ("Your request is still making its way through the bureaucracy").
By as a preposition has many definitions; the ones that interest us all loosely boil down to "using the means, mechanism, or agency of." If means or mechanism don't sound very different from system or process to you, don't feel bad. These are two of the more difficult prepositions to get right, and sometimes there is no right answer and you just use the one that feels right (or less wrong, anyway). Writers and editors argue over this and related matters all the time.
As a very general rule of thumb, if literal or figurative motion or progression is involved, use through. Otherwise, use by. There are tons of exceptions, however, and sometimes you just use whichever preposition is most conventionally used with the word or phrase, without asking questions.
To look at your examples specifically:
- My house is heated by/through gas. (Gas is the means used to heat your house.)
- I'll pay you by/through check. (A check is the mechanism you use to pay.)
- I learned that by/through a lot of practice. (Practice is a process, and you progressed through it from the point at which you did not possess the skill to the point at which you did.)
- Success only happens by/through hard work. (Hard work is a process, etc.)
- The tickets are available by/through the Internet. ("Through" is a bit of an evolved convention here. Think of the Internet in the popular 1990s-era conception of a virtual realm through which you travel to your destination.)
Those are the easy ones. The other two are harder:
- I'll send it to you by/through mail. (This depends on whether you consider "mail" to be a mechanism or a system. Generally, I would either use by mail or through the mail, with the idea that "mail" is a mechanism, but "the mail" refers to the postal system.)
- Through/by including a warning prompt prior to asking a SA related question, SPAM probes have been thought to isolate workload from the assessment of SA. (ugh, this is just an ugly sentence to begin with; I'd use by including or through the inclusion of, with similar reasoning to the above.)
These sentences both have perfectly acceptable grammar, but the intended meaning is not entirely clear. Consider this phrasing:
I have some long-lasting memories in my mind, such as memorable overseas trips that I have taken.
The phrases "been through" and "gone through" both can be used to imply enduring hardship or undergoing stress. People use them in sayings like "He looks like he's been through hell" or "I can't believe I had to go through that" or "Don't make me go through that again." This puts a negative connotation on the phrases.
If the overseas trips were negative experiences, if they went badly and are not fond memories, that phrasing is perfectly suitable.
If the overseas trips with enjoyable experiences, saying you "took" a trip is more common and clear.
Best Answer
Either is acceptable. The context would perhaps vary: if I was expecting a payment, and it arrived, I might say 'The payment has come through'. If I was sending or making a payment, and I saw it had left my account, I might say it has 'gone through'. But a recipient might also use 'gone through'.