Can I download a paid app on a second device from Google Play? Say I have bought an app from Google Play and already have it installed on one device. 

My question is – do I have to pay again if I download this app from Google Play from another device(but same Google account)?
Yes, you can have your Google account on several devices and you'll be able to use all paid apps on all those devices. There is a device limit, however (around 10 I think) which is meant to prevent abuse of that.