Are all the courts accross the world connected?

no they aren't connected in that way, not even in this country.

If you were to go on a multistate shoplifting spree, you can be charged with shoplifting in every state that you were shoplifting in. You would also be able to serve time for those crimes. It would be up to the particular court if they agreed to let you serve a concurrent or consecutive term (that is: serve the terms together or one after the other--say you got 10 years in one state and 5 years in another then you may be able to just serve 10 years and have 5 of your ten count for both states OR you may have to serve 15 years). Some states will allow the state with the stricter sentencing to handle the sentencing and will pass on trying you. (for example if one of the states decided that you deserved life for shoplifting, then the other states may drop their charges against you since you won't be getting out.)

Now you throw international law in the mix and again depending on where it is, they would either let the court with the stonger case and stricter sentencing take you OR they could decided to try you for the crimes you committed in their country and make you start your sentencing when you finished your sentencing for the crimes you committed elsewhere.

It can get very confusing fast.

Best bet, don't commit any crimes.

edit: as the previous poster pointed out many other countries will not send someone back to the US or anywhere else if they will face the death penalty.

There's complicated arrangments between countries called "extradition treaties" which concern people who have committed a crime in one country, but is in another. In general, if you committed a crime in country A, flee to county B, and what you did is also a crime in country B, country B will send you back to country A.

Note that many countries will not send accused murderers back to the United States, if the death penalty is a possibility, because they oppose the death penalty.

Also, if someone committed crimes in both countries, they will occasionally aruge over who gets him first.

No they aren't, because there are different laws in every country.