Will the last login be seen in WhatsApp if I’m not login for a month

On Android running Marshmallow, you'll want to disable doze for WhatsApp.

For stock Android, this link shows how to do that: http://www.greenbot.com/article/2993199/android/how-to-turn-off-doze-mode-for-specific-apps-in-android-marshmallow.html

On my LG phone, I went to settings -> General Tab at the top -> Battery & power saving -> Battery Usage

Scroll down till you see Battery Optimization and under that, tap 'Ignore optimizations'

Scroll down until you see WhatsApp and move the slider so it is on.

The messages are kept on the server for around 30 days, and after around half a year you're removed from your groups.

My brother was abroad for half a year, and since he didn't had internet most of the time he deleted whatsapp from his phone at that time.
When he installed whatsapp after a bit more than half a year, he was removed from all of his groups.

Note: If you don't have internet for lets say a few days, and then you connect and want to receive all the messages from your groups - the messages and media would probably be out of order, and you won't be able to understand what was going on...
I had that problem after a week abroad without internet - when I came back all the messages in my groups were disorganized and incoherent.

I know it's been a lot of time since the question was asked, but it can still be useful