Congratulations on your choice of holiday destination you definitely wont be disappointed. To answer your questions
1 There is no direct shuttle transfer from Linate the silvestri bus goes from Malpensa or Bergamo. You could arrange a taxi or hire a car. For taxi hire I can recommend www.valtellinaviaggi.com I am nearly sure there is a shuttle transfer between Linate and Malpensa but you would have to check that. Perhaps if you went on the Linate airport website you might get some information about this. It would of course depend on the arrival time of your flight whether you could get to Malpensa to link up with the shuttle service. You will need to book the shuttle in advance at www.silvestribus.it
2. Livigno is a long spread out resort made up of different areas. If shopping is high on your agenda you may be better off staying near the centre of the resort, although shops are open late in the evening (until 8pm)and there is a free bus service up til approx 7.30pm thereafter taxi is the best option and they are reasonable enough approx €8 per journey. We have been in Livigno every season for the last 6 (sometimes more than once) and always stay in San Rocco. This is a little further away from the centre but has excellent restaurants and bars and is near the lifts it is also near the one nightclub in Livigno the Kodoki. - although well spread out it is not a very big resort so everywhere is accesible.
3 See my sticky on skipasses. There is no benefit in purchasing online and ski passes can be bought from the ski pass offices of which there are many in resort. The free ski pass week's are exactly what they say - the lift pass is free is participating accomodations i.e. you ski for free.
4 I have very rarely heard a bad report on accomodation in Livigno - and my experience is that it is of a very high standard. You can view accomodation on the tourist office website www.livigno.eu or the ski pass website www.skipasslivigno.com they have accomodation to suit all budgets on there. If you put in your dates the apartment/hotel owners will come back to you with offers of accomodation.
5 In terms of access to the pistes Livigno is in the valley and there are two mountains one on either side of the valley so no matter where you stay within the town you are accessible to the piste. Most people ski the Carosello side in the morning as it has the sun and the Mottolino side in the afternoon as the sun is there at that time. There is a free ski bus, a lot of the accomodation is ski in ski out but if this is not the case the bus will get you on the piste within 15/20mins.
When you have selected your accomodation and know where you are staying come back and I should be able to recommend restaurants to you