european travel info needed...

my two best travel mates of all time are getting married. they're from stockholm, sweden. my sister might be going to usc in the fall so i want to take her on an all expense paid vacation for 3 months, i'm wanting to make this a two birds with one stone kinda thing. m y over all travel plan would be spend about a week in sweden for the wedding, visit another country or two in europe, then finally taking the transiberian train to beijing. from beijing i would work my way quickly down to hanoi, vietnam where i'll be meeting my sister at the airport. i've done most of south east asia so no tips needed there. but i have many questions as far as europe is concerned....
1) i will traveling under a US passport. will i need visas for any european countries, especially eastern europe if my stay per country would be less than a week?
2) i've been reading that trying to hop on the vodka train, ahem, i mean transiberian train from russia proves to be problematic. would it be so if i have a russian friend in moscow to help smooth things over, or am i better off hopping on in finland or germany?
3) will it be cheaper for me to fly to some other country, for example, england, then euro rail it to sweden or possibly fly? for the time being, a one way ticket from san francisco to sweden is $650 after taxes. if i'm lucky, i can find a flight from london to stockholm for about 30euros.
4) i know i'm missing a lot of questions, but if there's any info you guys think i should know. fire away!
thanks in advance!

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