If I pay for and download a game on my smart phone.......?

Its always with your google play account. So if change phones just go to google play store n download it again without paying nothing cause its already yours. Now if you ever delete your google email account by terminating it that you use for the google play then youd have to buy it again with your new email n account

Sign into the same account & check for it under a purchased section.

(Assuming you switch to a phone with the same OS)

Generally you just have to set the new phone up on the same account - which you should be doing anyhow! So if you've more than one device on the same account you'll normally have access to the game on those devices.

*This isn't always the case, but when it is not, you usually have the option to select which device on the account the game is available on!