As downloading without owning the rights equals stealing to some people, can an islamic court that bases it's laws on the sharia cut the stealing person's fingers?
Of course no. First of all, in Islam , to get your fingers ( actually palm , not fingers ) cut , there are conditions to be fulfilled first: one of them if that the value of the stolen thing is higher than a limit put by the court called "nysab", I am sure that this movie or whatever your downloading won't exceed it . Second of all, in my opinion , I don't think torrent or any cracked program is stealing . It's not considering stealing if you're not stealing from someone, which you're not , cause the up loader made it free to download, even if it is cracked. Hope this helps, thanks for asking .
An Islamic Country doesn't necessarily mean that it implements sharia wholistically, if that is what you are asking. Currently there is no country in the world where there is the existence of the sharia as a whole, countries like saudi arabia are monarchies with bits of sharia.
If you are talking about an Islamic State ie a state that bases its ideology on sharia entirely (Caliphate), then it wouldn't be that straight forward to cut peoples hands no! For example at the time of Umar's (2nd caliph of Islam) Rule he suspended the rule of cutting ones hand due to the state not being able to provide for its citizens.
For your question you would have to define if that constitutes a theft in the first place.
England was once a Christian country. At one point, not really that long ago, you could be hanged for stealing apples if you were a poor pauper. Of course they do not do this now.
Sharia is not a monolithic legal construct, there are different schools, and it will change with the times. The law of Allah, the formal law of the people, the law as it embodied in the people, and the law as it is executed are all very different things.