My upkkep begins, and I want to activate the tetravus' ability and put 3 tokens into play.
As far as I understand, energy flux's effect is static giving tetravus a triggered ability in the upkeep cost. So tetravus' upkeep cost-ability goes on the stack, I pay, and its resolved.
After this, if I then activate the tetravus and put 3 tokens into play, these tokens will all get the triggered ability, but will that ability go on the stack? Will I have to pay for the 3 tokens this upkeep too?
Tetravus has no activated abilities; rather it has two triggered abilities that trigger "at the beginning of your upkeep", just like the ability Energy Flux grants to "[a]ll artifacts". When your upkeep begins, all three abilities will trigger and go on the stack, and you can choose how to order them on the stack. The topmost spell or ability you put there will get to resolve first (C.R. 603.3b). For example, you can order those abilities in the following way:
Bottommost: "At the beginning of your upkeep, you may remove..."
"At the beginning of your upkeep, you may exile..."
Topmost: "At the beginning of your upkeep, sacrifice this artifact..."
And ability 3 will get to resolve first. Or you can order those abilities in the following way:
Bottommost: "At the beginning of your upkeep, sacrifice this artifact..."
"At the beginning of your upkeep, you may exile..."
Topmost: "At the beginning of your upkeep, you may remove..."
And ability 3 will get to resolve first.
Remember that you don't activate a triggered ability, even if it has the form "At the beginning ..., you may .... If you do...", or "At the beginning ..., [do something] unless you pay...". And you don't choose to pay for any of those abilities until that ability resolves (C.R. 118.12a, 603.5). See also this thread.
After this, if I then activate the tetravus and put 3 tokens into play, these tokens will all get the triggered ability, but will that ability go on the stack?
The tokens will be given "At the beginning of your upkeep, sacrifice this artifact..." due to Energy Flux, but the ability given to each such token won't trigger until the next time that token's controller's upkeep begins (C.R. 109.5). (By the time you create a token with Tetravus, it will happen during your upkeep, not at the beginning of it. Remember that the "beginning of your upkeep" is a single moment in time, at the point that your upkeep begins. See also this thread.)
Will I have to pay for the 3 tokens this upkeep too?
No, you don't pay the Energy Flux cost for the tokens since they didn't exist at the beginning of your upkeep.
I have a Tetravus with 3 counters on it and my opponent has an Energy flux.
My upkkep begins, and I want to activate the tetravus' ability and put 3 tokens into play.
As far as I understand, energy flux's effect is static giving tetravus a triggered ability in the upkeep cost. So tetravus' upkeep cost-ability goes on the stack, I pay, and its resolved.
After this, if I then activate the tetravus and put 3 tokens into play, these tokens will all get the triggered ability, but will that ability go on the stack? Will I have to pay for the 3 tokens this upkeep too?
Remember that you don't activate a triggered ability, even if it has the form "At the beginning ..., you may .... If you do...", or "At the beginning ..., [do something] unless you pay...". And you don't choose to pay for any of those abilities until that ability resolves (C.R. 118.12a, 603.5). See also this thread.
The tokens will be given "At the beginning of your upkeep, sacrifice this artifact..." due to Energy Flux, but the ability given to each such token won't trigger until the next time that token's controller's upkeep begins (C.R. 109.5). (By the time you create a token with Tetravus, it will happen during your upkeep, not at the beginning of it. Remember that the "beginning of your upkeep" is a single moment in time, at the point that your upkeep begins. See also this thread.)
No, you don't pay the Energy Flux cost for the tokens since they didn't exist at the beginning of your upkeep.