There are 268 miles from Barcelona to Avignon.
Yes, there are several autobuses from Barcelona to Avignon. Check companies Linebus, and Alsa.
The road distance from Paris to Avignon is 692 km. The travel time is about 6H30 by car, or 2H38 when using the TGV (high speed train link).
About 5 hours by train, according to this website: http://gospain.about.com/od/bustravelinspain/a/madrid_bustrain.htm
Barcelona has a train station, so it is possible to get there by train. However. where you are starting from (which you have not specified) may not have a station or indeed be connected to a rail network.
22h10 including night train (sleeperettes) between Barcelona and Milan.
To be honest, it's probably better to fly, but if you want to take the train, you will need to take a train first from Barcelona to Venice in Italy. Then you will need to take the train from Venice to Vienna.
The trip will be approximately 18 to 22 hours.
There is no high-speed train connecting Lisbon and Barcelona. The high-speed train from Barcelona to Madrid takes about 2hr and 40mins. The normal speed train from Madrid to Lisbon only leaves once a day and takes about 10hr to get from Madrid to Lisbon. It is a smarter idea to take the high-speed train to Madrid and transfer to a bus that will go to Lisbon.
Not directly, you'll have to change twice.
Yes. By bus, train or plane.
By train - overnight train 'Trenhotel' leaves Barcelona Estacion Del Norte at 9pm and arrives Paris Austerlitz station at 9am (making 4 short stops en route) By plane - depending on which airport you fly to: Barcelona to Paris Orly 1hr 45 mins Barcelona to Paris Charles de Gaulle 1 hr 55 mins