First problem

Google treats the retrieving of external POP3 accounts using their own algorithm and that says in sum that, if you get many email, it will fetch with small gap times, if you have emails once upon a time it will expand the recurring time as there is no need.

You should avoid this completely if you need that POP3 emails quick, for that, you need to create a simple FORWARD email that what it does is

every time I receive an email, I will forward to the address xpto

from your Control Panel, just do this:

This will make sure that you do receive as fast as you can (as it's at the same time it reaches the POP3 server) and if you configure your GMAIL to filter all emails that were sent to your whateverdomain.com you can easily flag them and even forward again, but this time using GMAIL.

Second problem

You can have multiple accounts on your iPhone, I have several as the image below shows:

Just add your Gmail and your POP3 Account.

Third problem

If you use a Forward account, you cant set up a new POP3 account, as it's not a POP3 account anymore and you are not keeping anything on whateverdomain.com just in your forward email.

If you need to keep things in the POP3 Account as well, I would suggest that you create a new POP3 account and on GMAIL (that receives the forwarded Email), create a filter to Forward it again to the new POP3 account (or a new GMAIL account just for this, for example as the basic storage size is always a little bigger than regular POP3 account sizes).

Something like this:

Then in your Mobile phone you can use only GMAIL (as the messages are being dropped into your personal account) or set up the POP3 account and you have one account to send and receive emails just for the whateverdomain.com

The shortcut that does exactly what is asked for here is 'u', which Google labels "return to thread list," which is what it does if you're in a conversation view (i.e. a single email thread). However, if you're already looking at a list of threads (e.g. your inbox), 'u' will refresh that list.

Unlike "g, i", 'u' works to refresh any thread list you're looking at, and also does not move the cursor back to the top.

Like "g, i", the 'u' shortcut is part of the set of keyboard shortcuts that might be disabled. If that's the case, you can enable them by setting "Keyboard shortcuts on" in Gmail settings, under the "General" tab. You can also do it by pressing the '?' key in the thread or conversation view, which brings up a list of keyboard shortcuts and their meanings (although, as demonstrated here, those meanings aren't always clear). There's a link in the middle of the box allowing you to toggle whether the extended keyboard shortcuts are enabled or disabled: