Prepositions are perhaps the most polysemous of words. And all words are polysemous.
It's tempting to say 'You just have to learn every individual collocation'. So don't start teaching until you're past retirement age (by which time usages will have changed anyway, so you'll have to go on a refresher course ...).
As jimsug says, there are instances where the two prepositions are definitely not interchangeable; often, they are highly idiomatic:
I learnt it by heart/rote.
*I learnt it through heart/rote.
.............
*She stuck with him by thick and thin.
She stuck with him through thick and thin.
Though they may be central senses (directional / locative):
I drove by the site of the old Chicago Motor Speedway. =/=
I drove through the site of the old Chicago Motor Speedway.
In 'they will do this by / through a detailed research project', the more peripheral sense/s of agency/means are in play:
I could tell it was you by what you were wearing.
He made it by hand.
We went by car.
He only discovered this by chance / through good fortune [far more common than the
next alternative ?!] / by good fortune.
You will only succeed through hard work / by working hard.
'They will do this by a detailed research project' is possibly best considered as an ellipted form of
'They will do this by carrying out a detailed research project' or
'They will do this by means of a detailed research project' or
'They will do this through the expediency of carrying out a detailed
research project' or
'They will do this through the expediency of a detailed research
project'.
This is perhaps an illustration of part of the evolutionary processes that can be involved.
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'.
Best Answer
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:
Those are the easy ones. The other two are harder: