Mail is delivered as soon as it arrives at gmail server instead of every hour as I requested

Question

I have set my mail preferences to get mail every hour. But mail arrives when it arrives. It can be every few minutes. This has been going on for a while (Not helpful, I know, but bear with me.)
For the last few weeks I have been getting an exclamation mark next to one or the other of my gmail accounts with errors that range from 'non secure device' to ' too many connections'. Trawling through the gmail forum has led me to follow all of the steps they suggest but the 'too many connections' things won't go away.
One point they make is that if you are 'getting' mail more frequently than every 10/15 minutes this can cause the problem.
Now, if I could pinpoint the actual time when this phenomenon of mail appearing when it wants to began, we would have an idea of whether it was connected to the latest Yosemite update. Unfortunately, I can't.
But I have trawled all through the account settings on google and cannot find any place where I set when gmail sends my mail so I reluctantly conclude that Apple Mail is ignoring my setting of 'check once an hour' and is checking at some randomly selected period.
Does anyone else have this problem? Does anyone have any idea about how to fix it. I am fed up with going to gmail captcha and telling it again and again to accept this device; with signing in on gmail to the account in question; with this whole stupidity in general. Once I do all of the above, everything seems fine until the next mail arrives when the exclamation mark appears again and I have to go through the whole rigamarole again.
Thanks for any help.

Answer 1

Thanks for the detailed procedure, it helped a lot!