Well done on holding out against an attempt to foist unwanted services on you. Sky cannot put you on a huge package by default, as a contract can only be formed by one party accepting the other party's offer unconditionally.
There were terms in your O2 contract allowing transfer of O2's obligations to a new party, and to allow new terms and conditions to be imposed with due notice and after giving you an opportunity to leave. However, this does not give Sky carte blanche to do as they please with the transferred customers.
What's more important for you now is the issue with BT. You will still have the same line, but it is now connected to different equipment at the exchange.
Please post your router statistics (attenuation, noise margin, anything else you can access about errors and the like), or the make and model of the router, so that someone can help you find this information.
If you have any statistics (not just the speeds) from your old O2 connection, that would be a useful comparison.