I don't think todays west is Edom. Edom was Eisav, Christians are descendants of Yaakov. It might be attractive to liken todays current events to what we know about Moshiach, but don't forget R Akiva said that almost 2,000 years ago.
The culture and continuation of Edom is deffinitly "the west". The kingdom of Edom itself has been destroyed.
Where did you get that, that today's christians are descendants of Yaakov? Christians are made up of many many many goups. I beleive there is a mahrsha in chagigah or taanis (I forget) on the gemora "garmamya shel Edom" that says it's germany. The original christians went begging for followers and brought in many pagan cults, hence december 25th with it's trees etc;.
R' Akiva thought in the beginning that Bar Kochba was moshiach, but soon after realized he deffinitly was not.