>>> I take it all the e mails 'grabbed' by gmx will be safe
There about as safe as it's possible to be. GMX is run by the United Internet Group. They also trade as '1&1, managing over 10 million internet domains, including those of many multinational companies. They're very much 'in the big league' when it comes to email providers (although, surprisingly, not that well known in the UK). All of your data should be safe with them.
>>> when I have tried Outlook they where there also.I'm not sure exactly what this means,will they be gone or not?
The original way that email clients (such as Outlook) worked was to connect to your email provider's server and download all of your mail onto your computer. By default that mail was then erased from the provider's server (but you could change the settings in the email client to tell it to leave a copy on the server if you wanted to). That was a using protocol known as 'POP' (or 'POP3').
However many email services now offer IMAP access to your mail (either as well as, or instead of POP). With that system your mail is stored on email provider's server but accessed through your email client. (Think of it as a sort of advanced webmail service). However it should be possible to change the client's settings to keep a copy locally.
So, depending upon whether you've been accessing your email with Outlook via POP or IMAP (and possibly also upon the settings within Outlook) your mail might be on your computer or on Virgin's server. The simplest way to check is to disconnect your computer from the internet and see whether your mail is still available through Outlook. (If it is, it must be stored on your computer. If it isn't, it must only be on Virgin's server).
>>>My main worry now is if I where using a website (ie theanswerbank) and I forgot my password,and the website kindly offered to send my password to my e mail address,I would be in trouble wouldn't I?Can anything be done about that?
You can, of course, change your registered email address with each website you use. (On AB you go to 'My Profile' and then to 'Edit my Profile'). However, as someone recently pointed out on the radio, the chances of someone breaking into your house and stealing the notebook that you've written all of your passwords down in (and then going on to use them maliciously) is actually incredibly low. Simply writing them down somewhere is often the best way to go.