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

QUESTION 1-In the case of Microsoft Office Outlook Web Access, it does not look possible for gmail to access them. Gmail appears to need POP available, and my administrator says that without IMAP or better it is a waste of time. So you have to use microsoft explorer to access the web mail for the full set of options to appear and then set up a rule for forwarding or redirecting email to an address (instructions).

QUESTION 2-There are filters to add tags for organising the mails according to the to/from (etc) features or the emails.

QUESTION 3-Yes

QUESTION 4

1. Top right click on the cogwheel and on the parts that open below click on account settings
2. Below the word SETTINGS, click on the tab 'Accounts and Settings'
3. The section 'Send mail as' is the relevant section
4. click on 'Add another email address you own'
5. input the name you have on registered on your other email account (non-gmail) and the email address and click next
6. click on 'Send through strath.ac.uk SMTP servers'
7. you will need to put the username and password you use to access the non-gmail account and the SMTP Server address which may not be the default. The smtp address is usuall known to you if you set it up, or ask the admin somehow or the institution that set it up might have a webpage listing it, search it. I have left the port and security connection as default.
8. If the information is correct it should work, and when you send an email for a new composition there is a list of addresses you can choose from, the default and the new one. When the new one is chosen the previous/default is not visible.

Google help here for question 3 and 4.