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"As I understand it, no process can happen if it's not increasing the overall entropy of a system, therefore the universe is always going towards more entropy."

Not exactly, this law is of statistical nature and it is generalized on the basis of experiments that we can observe on Earth. Some people argue that on a bigger scale entropy can decrease, i.e. in the stars/galaxy formation, etc.

'And as I understand that, in a system with total entropy, there's no time.'

I've never heard about that, source? As far as I know time has no connection with entropy.

'Now my question is, is this change towards entropy slowing, accelerating, or neither?'

I doubt that anyone can tell. I don't know.


'Put another way, does the 2nd law imply that the universe will at a certain point reach total entropy, or could it be the case that even though it has more entropy as time progresses, it can never reach a state of total entropy since this change is being slowed by orders of magnitude?'

I don't know. I doubt that it is possible to reach the state of 'total entropy', for example if the universe will continue to expand, it is impossible to reach the state of total entropy. (This state would look like if all matter is dispersed uniformly all over the universe, it doesn't seem feasible with expanding universe.)


'A second question - if we can choose what process to let happen (say use an engine vs using horsepower, etc), does that mean we get to choose to some extent the speed at which the entropy of a system progresses to?'

Yes we can but it will have a negligible influence in comparison to increase of entropy brought about by the reaction in the sun.

'
And a third, related one: If a system has an X "level" of entropy, does that mean that in average, all processes will yield Y progress in entropy? In other words, given a set amount of entropy, is the average speed at which processes produce entropy fixed, or can it vary?'

No, it is not fixed. It depends on the specifics of the process.



'Thanks a lot for taking the time to read (and hopefully answer!) these questions. If you want to add any additional information that you consider relevant, I'll be grateful for that aswell.'

You should keep in mind the statistical nature of the 2nd law of thermodynamics and probably check the statistical mechanics foundations of this law.