Learn Croatian: Tko je izmislio taj stroj? - Who invented that machine?

The base verb is misliti (to think). The prefix iz- adds a sense of bringing an action to its ultimate conclusion or origin. Here, thinking something all the way through until a new concept exists creates izmisliti, which means 'to invent' or 'to make up'.

Since taj stroj is the object of the sentence, why isn't it in the accusative form tog stroja?

It actually is in the accusative case! Stroj is a masculine noun. For masculine nouns in Croatian, the accusative case perfectly matches the nominative case if the object is inanimate (not alive). Since a machine is inanimate, it stays taj stroj. If it were a living being, it would take the animate accusative ending, which looks exactly like the genitive form.

What is the imperfective version of izmisliti and when would I use it?

The imperfective form is izmišljati. You would use it to describe an ongoing, repeated, or habitual action, such as 'He is always making things up' (On stalno izmišlja). In this sentence, the invention of the machine is a single, completed event in the past, so we need the perfective izmislio.

Does izmisliti only refer to inventing physical things like machines?