Learn English – How to define “by” versus “through” to the students

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.)

The semantic class of adversatives—words that, as you put it, “set up relationships of contrast,” or of dialectical tension—overlaps various grammatical classes (parts of speech) including preposition despite, subordinating conjunctions although, though, while, etc., adverbs though, however, nonetheless, etc., and coordinating conjunction but. See this previous question: I had not thought to include despite when I answered that, but have now repaired the omission.