if the screen is broken can i check my messages and missed phone calls

While it won't help you check SMS message, you can check your voicemail without the iPhone. Contact you carrier and find out the sequence necessary for you to access your voicemail from any other phone.

Well if the screen is broken, you can't very well do that, can you? Bring it to Apple for out of warranty replacement or find a 3rd party repair shop near you.