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December is not a proper winter. No snow at the moment, may be some snow soon, but it won't be suitable for any kind of skiing. Winter may turn into its proper stage since the second half of December, but equally may turn not.
One way to Kaunas (from Vilnius) is not nice (A1), another way is nice (A16).
Drive to Daugavpils is not scenic, the scenic one is road A16 towards Kaunas, but the way to Daugavpils is longer than twice so it's rather diverse, so there are rather nice parts and rather dull parts. Those "nice parts" now are not exactly nice because it's the end of autumn/beginning of winter now. It's not a nice time of year so there can't be scenic drives or nice roads by definition. To be more specific - the end of November and the beginning of December is the ugliest and worst time of year, that means that some time earlier and some time later is somewhat nicer and somewhat better.

"...What is the best advice for a daytrip by car from Vilnius..."
I suppose that "Daugavpils option" is not a "day trip"? Nobody go to Daugavpils specifically as all objects of interests, all recreation areas and resorts are between Vilnius and Daugavpils (no matter whether you drive down the A14 or No102), Daugavpils may be a landmark, sort of the "end point" where you turn back or so, but not a destination. Unless you have a personal interest, then it's another story. As for those "objects of interests in between" - they all are away from the main road, one is not supposed to enjoy driving between Vilnius and Daugavpils: one is supposed to turn from the main road to the secondary road, to another secondary road and so on, according to ones approximate plan what he wants to see and where he wants to go. For example, when you turn from the main road to the secondary road, that secondary road is supposed to look, e.g. like this one http://foto.terpe.lt/inkelti/20110506/i74_VK4406%20Sariai-Svencnl%202011%2004-01.jpg , and another secondary road - http://img2.oplius.lt/beta.plius.lt/images/anns/domoplius/5/16762/8/parduodamas-sklypas-ignalinos-r-sav-ginuciuos_3170520_normal.jpg , and, for example, you arrive to this or similar lake http://www.ramsar.org/pictures_2009/Country/lithuania-labanoroparkas.lt.jpg , lake shore http://ezerai.vilnius21.lt/nuotraukos/5539d.jpg (there are few hundreds of lakes on the way to Daugavpils, one can visit several dozens in several hours so recommendations are unnecessary), you pass, e.g. this or similar village, visit this or similar object http://safari.lt/uploads/images/P7013898.JPG , climb this or another hill fort http://foto.terpe.lt/inkelti/20090912/i16_Piliakalnis_3.jpg , and so on. If you intend to go specifically to Daugavpils for some reason and return back same day, then you have to go directly there and skip everything in between, it's not scenic trip, it's a trip to Daugavpils specificaly.

Road A16 is scenic in itself, secondary roads (as described above) would take you to the certain objects, all surroundings are rather scenic. To go towards Kaunas is OK even if you don't reach Kaunas itself. Depends on what kind of day it will be, sunny or rainy.

Given the time of year, i'd say that "the best advice for a daytrip by car from Vilnius" is a trip to Kernavė.
If one prefer more intense trip, one can go to Kernavė via Šilėnai village, there is a tiny winding road (suitable for cars) across the forest. You turn from the main road, drive some 1 hour (with or without short stops) and finally return to the main road but closer to Kernavė and proceed on. This is best. Others are worse. To my taste, of course.

Belarus border question is probably not for me.