As for the oil burning, did it have a lot of blowby? Sometimes a lot of oil burning can be attributable to a broken compression ring-which in turn can cause damage to the oil ring-causing some smoke. To eliminate this possibility-once it is reassembled do a compression test on each cylinder
Another way the oil can get in is via the stem seals (as you said) A good way to check is to have a look at the valve shafts through the intake/exhaust ports. If the shaft of the valve looks all oily and the back of the valve has a carbon built up on it-looking like it has run down the shaft of the valve, then the stem seals are shot.
As for the movement of the pistons-a description like that is indicative of one of three things generally
1. The pistons installed were too small for the bore-ie 10 thou oversize pistons in a 20 thou over bore. (not all that likely-its a pretty big screwup) 2. The pistons and bore are totally flogged out and need attention. 3. The motor has been fitted with big clearance race style forged pistons (ie my Venolias could be wiggled an alarming amount in the bore.) The big clearance is generally because a high silicon content material with a big thermal expansion property has been used.
Have a look at the markings on the piston crowns and ask a lanci/alfa workshop what type of slugs they are and what the clearance is supposed to be.
Have the radiator checked as well-having it opened up and cleaned out at the local radiator shop is cheap insurance.
It could be a head gasket-but if the gasket looks okay between the flamerings in the gasket and the cooling passages, then it probably is okay. It could be a hole in the head somewhere, so have it checked for porosity (when you have it skimmed) Also have the head flushed out.
I'd recommend the following
Keep the gasket you have-don't but a new one yet-you cant re-use them reliably-but it'll be fine to reassemble the motor and diagnose with-just clean the head and block up carefully with a razor blade and scotchbrite pads. Next get a big metal ruler or similar and check the head for flatness. (lay the edge across the head in a few directions-if it rocks at all-have the head skimmed-do the same for the block). Get a can of Hylomar or similar head gasket adhesive spray and put the head back on (change the stem seals while you're at it) Do a compression test, then see if it still behaves wierd.
If you've go any Q's about what I've said just post again-No question is too basic to ask-don't be shy.