I have tried to sign in with Windows Live ID but it didn't confirm as it's okay or not. Also, I want to install applications but I can't. Why is this? I'm using a Nokia Lumia 610.
You are using a Nokia Lumia 610 and WP7.x.
As of the official Windows Phone 7 support page:
Windows Phone 7 has reached end of support, so services and functionality for this version of Windows will be discontinued over time.
** Download apps and games FAQ,
You may not have the latest version of the Windows Phone operating system (OS) on your phone. You'll need Windows Phone 7.5 to download and update apps.