You can use the Sky dish to connect to a Freesat box.
Connecting a Freesat box will not clash with a Freeview television. You just connect the box using a SCART lead, or HDMI (if it's an HD Freesat box), to the telly. The you just select the appropriate AV input on the television and use the Freesat box as your "tuner".
If your new Freesat box has a built in HDD recorder then use the box controls to record and play programmes. This will mean very carefully reading the instructions - some of them are VERY badly written (Korean/Japanese-American-ish) so be prepared to be patient.
Good manufaturers (eg Panasonic) go to the trouble of providing decent English instructions with their products. This makes using them very much easier.
The important part of the satellite dish is the Low Noise Block (LNB). This is the unit mounted in the middle of the dish, it does the actual reception of the satellite signal. The dish itself is just a reflector to help capture and focus the signal. If your installation is very old or damaged then the LNB may be broken. You can buy new ones from www.maplin.co.uk for around £20 or less.
Installation of a new LNB is just a matter of unscrewing the mounting bracket that holds it to the dish and swapping for the new one. You do not need to remove the dish from the wall, but be careful not to knock it out of alignment.
The plug is an "F Type" connector. You can buy these from Maplin too. They're easy, but fiddly, to fit to the coaxial cable. You need dexterous fingers.