Cannot setup Mail to my fasthosts account? Please help

You can turn your Gmail account off.

Go to Settings > Mail, Contacts, Calendars. Select your Gmail account. At the top, there should be an account on/off selection.

If you need to leave your Gmail account on, any messages that remain on the server will continue downloading. The 25 email settings is a show emails in a mailbox setting. It does not prevent new or additional messages from downloading if there are additional messages available on the server in the selected mailbox.

Doubtful your fasthosts account can be accessed as an Exchange account via ActiveSync, which is how the account is accessed when selecting the Exchange account preset when creating the account. More than likely the account is accessed as a POP account.

Correct.

If you used the Gmail account preset when creating the account, the account was created as an IMAP account. All messages in all server stored mailboxes will be available when accessing the account with the iPhone's mail client, which would be the same when accessing the account as an IMAP account with an email client on your computer. With show the 25 most recent messages in a mailbox selected, when deleting the existing 25 most recent messages shown in a mailbox will show the next 25 most recent messages if there are more than 25 messages in a server stored mailbox.

If you create the account as a POP account, all messages that remain in the account's Inbox mailbox at the server will be downloaded as new and unread messages even if already read on the server or read with another email client used to access the account. The same would apply when creating the account as a POP account with another email client on your computer or with an email client on a new computer.

I'm not sure what the issue is. The idea of accessing an account as an IMAP account with any email client is having access to all server stored mailboxes and messages, with the ability to keep all server stored mailboxes synchronized with the server automatically with each email client used to access the account.

Is the iPhone's Mail client the first email client you have used to access the account - you have always accessed the account via webmail using a browser?

The iPhone's Mail client is functioning as designed - as every email client used to access an email account is designed.

If you are accessing the account as an IMAP account with the iPhone's Mail client, all server stored mailboxes and messages within those mailboxes will be available - as intended and designed with any email client used to access the account as an IMAP account.

If you are accessing the account as a POP account with the iPhone's Mail client, only messages in the account's Inbox mailbox on the server will be downloaded as new and unread messages - which would be the same when creating the account as a POP account with any email client.

If you don't want any of the existing messages available on the server available when accessing the account with the iPhone's Mail client, you need to remove all messages from all server stored mailboxes on the server if you are accessing the account as an IMAP account. If you are accessing the account as a POP account, you need to remove all messages from the account's Inbox mailbox at the server if you don't want any of the existing messages in the account's Inbox mailbox at the sever downloaded by the iPhone's Mail client.

The same applies when accessing an IMAP account with any email client, and the same applies when creating a POP account with any email client in regards to any recieved messages that remain on the server in the account's Inbox mailbox.

With an IMAP account, the idea is to have access to all server stored mailboxes and messsages with each email client used to access the account with all server stored mailboxes for the account kept synchronized with the server automatically with each email client used to access the account.

This is what an IMAP account is specifically designed for.

When creating an existing POP account that is being accessed with one email client with another email client on a different computer or with the iPhone, all recieved messages that remain in the account's Inbox mailbox will be downloaded as new/unread messages with the new email client being used to access the account - even if the messages have already been downloaded and read with another email client. The account's Inbox mailbox with a POP account is not synchronized with the server with each email client used to access the account. This is the nature of a POP account, which cannot be changed.

The iPhone's Mail client is working as designed and intended, as every email client used to access an IMAP or POP account is designed and intended.