What computers or electronics things to bring to live?

I am not sure if the cable modem you are bringing will necessarily work for the local cable internet company. It isn't a big device so no harm in trying it. Keep in mind that while cable modems are fantastic and the best way to go in the US, they are normally not the best way to go here. DSL is often superior.

I see no reason for a separate LCD monitor unless the laptops are not sufficient. Better to bring an LED TV than a monitor.

See you soon..

Yes, power strips/surge protectors. I dont know the power situation in your house but you dont want your power supply to break due to a power surge. (ditto for all appliances)

if you have the times in your list on you and have space in your baggage bring it with you. It will save you the trip to a mall.

Bringing in routers will help as every building is built with concrete so Wi-Fi signal doesn't travel far. And I would bring in extra cheap LED TV from Amazon, so you have backup and use that as your monitor.

Think about what you will need and bring EVERY THING you need and want. It will save you trip and time.

I work remotely from down here and all I have that I couldn't easily have gotten here is my laptop and Vonage box. Of course you can get a laptop here, but it would cost and you wouldn't get the one you want with your stuff on it. You could get a used Vonage box here from someone wanting to sell. You can get a router here, but you will pay more so bring yours if you can. I doubt you will be able to use your cable modem here. You can get cheap phones here (house phones like the one I connect to my Vonage box). An actual Wi-Fi extender might be good if you are going to eventually move to a big house. All of the stuff you are talking about you can find here pretty easily. Some may cost a few bucks more, but don't kill yourself by paying a ton of extra shipping costs or checking extra luggage that you have to pay for and haul around.