LR
I am afraid your posts do not make it very clear what problem you have.
Hotmail is a web based e-mail service. You log on to Hotmail via an internet browser programme, enter your e-mail address and password and all your e-mails and contacts are there to see. You can use an e-mail client (e.g. Outlook) to view and write e-mails but it is basically a mirror image of what is seen on the web portal.
BT will have had nothing to do with any problem with Hotmail. If you are using Internet Explorer has your Menu Bar or Favourites Bar disappeared, if so it is just a question of selecting them again? The method depends on which version you are using. E.g. in IE9 just right click alongside one of the tabs and a menu will apear where you can select the missing items.
Regarding speed most (I think) ADSL connections are 'rate adaptive'. That means that a new connection will over a period of around two weeks try to maintain a connection at the highest sustainable rate. In practise this means the connection may be unreliable and drop frequently as the equipment attempts higher but unsustainable speeds. When it settles down it will be at a rate which is the best it can sustain with any accuracy.
Hope this helps.