Learn English – Next week, in the next week, for the next week

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.)

It depends what you mean.

"I'd like to schedule a meeting with you next week" could mean that sometime next week, you'd like to schedule a meeting. That meeting might actually be scheduled for next month, but next week you'll agree to the time.

"I'd like to schedule a meeting with you for next week" means that the meeting should take place next week.

Edit: "I'd like to schedule a meeting with you for next week" could also mean that you want to schedule a meeting to talk about something that will happen next week. The actual meeting will happen this week, and you want to talk about next week's event.

That's just one literal interpretation, and the context of where and when you're saying it might make it more obvious that you mean something else.

Edit: You've just changed your question, so I'll add this:

"I'd like to schedule a meeting with you in the next week" means that sometime in the next week (which includes whatever is left of this week), you'd like to schedule a meeting. That meeting might actually take place next month.