-Visual Voice Mail is carrier dependent, so that's up to the carrier you're on (most charge extra for it, though). If you switch your number to a Google Voice number, however, you get visual VM for free, but since it's only a Google Voice number, you don't get the same terms as your carrier (unlimited voice/text to US and Canada, but international will cost... low rates, but not included as it is with some carriers now).
-Any other instant messaging app like Whatsapp, Facebook Messenger, Telegram, WeChat, etc. If you just want the ability to text from your computer or other device like you can with iMessage, then you can do that with apps like Push Bullet or, an all-time favorite, MightyText (most likely what you'll like). If you are on T-Mobile, you can activate your line as a DIGITS line and be able to text/call and sync that on any device and/or computer. Or, a third option, is again, to switch your number to a Google Voice account and be able to text from Hangouts on any device or web browser. Also, some manufacturers, like Samsung, have their own options for text sync with a computer (i.e. Samsung's Flow)