In Gmail you can use a search operator which if you get more than one different email address forward to you in the search bar you can typpe to:theemailaddress@gmail.com and then you can get all that mail from that address, you can also do that for mail from a address.
With a IMAP client you can index the files better which means more faster searches.
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
Best Answer
You have two options:
There is a thread on Gmail help dealing with this exact issue: http://www.google.com/support/forum/p/gmail/thread?tid=2d69c50302cf0e14&hl=en