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

This is a known issue with all of Google's services. They do not support the .eml or the less common .mht, both of which are known and established formats. In this case it has been (for years now) one area that all the others (Microsoft, Apple, Mozilla, and most of the rest of the community) are ahead of Google on. The only option, aside from convincing Google to support it, appears to be to either bear with the extra info and read it as a raw text file or to save it and open it with almost any other email client, such as Thunderbird or Outlook.