I need to drive from Calgary to Seattle, WA, this month (March), but is worried about driving through the mountains in winter along Tanscanada Highway #1 (icy lanes, avalanches, etc.)

Will it be safer to take highway #3 via Vancouver to Seattle?

Or even going futher south to Montana and take US I-90?

Here's a website that has road conditions for BC and Alberta:

http://www.highwaylodge.com/reports/roads_ab.htm

March is a very variable month when it comes to weather, so it is really difficult to predict which route is best. And it changes from day to day.

However, the Trans Canada is a much better highway than highway 3, and has lots of semi (truck) traffic on it.
When travelling from Vancouver to Kamloops, drive the Coquihalla toll highway, and then take the Trans Canada to Revelstoke, Golden and on to Calgary.

Plan to travel more slowly than you would in summer, and make sure that you have good tires (chains may be required in some sections of the mountains).

I don't know whether you would avoid snow and ice by driving through Montana; the high mountain passes in western Montana probably have the same sorts of late winter-early spring road conditions that the Rockies in Canada do.