Yes and no.
Only if the device is the same family as the original account. If you have a computer minecraft account and want to play Minecraft on an X-Box you won't be able to.
X-box Minecraft accounts will not work on an iOS Minecraft account, etc.
You still need to buy seperate iOS, Android, X-Box, Playstation and Computer games/accounts. (Mac, Windows and Linux can use the same account tho.)
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You should be able tto download it as soon as you've paid for it.
It is illegal to play Minecraft without an activated premium account, and WikiAnswers will not provide guidance for illegal actions.
In a sense, yes, but in another sense, no. You do not need to install Minecraft to play it on a computer, you can simply run it in your internet browser on Minecraft.net. However, if you want to run Minecraft without an internet connection, or without visiting Minecraft.net, you must install Minecraft on your computer. You can install Minecraft on multiple computers under the same account, but your saved worlds will not sync between computers.
Check out the link below for a guide of how to install mods in Minecraft.
I think either because you did not install the minecraft properly or you typo error at your password or your username