Can the Bering Sea Rule be used to predict the weather?
There are two papers produced by the NOAA's National Weather Service investigating the claims regarding the Bering Sea Rule (BSR) and the Typhoon Rules for long range forecasts.  Some insite for your immediate questions:


Although not giving a percent reliability of the BSR, the second
study does say that the BSR has forecast skill above climatology for
long range forecasting - see Discussion, Summary and Conclusions
section of study II.
From paper I: 


  The Bering Sea (BSR) and Typhoon (TR) rules are two observations used
  by weather forecasters. The former was introduced in 2011, while the
  latter has been used since at least the 1940s and are based on the
  idea of teleconnectivity within the Pacific Ocean region which was
  defined statistically by Wallace and Gutzler (1981). For example, the
  Pacific North American (PNA) pattern will be associated with
  alternating trough-ridge patterns from the Central Pacific to the East
  Coast of the US. A ridge-trough pattern from west to east over the US
  is a positive PNA configuration, while the opposite pattern is
  negative. Teleconnection is thought to be the result of downstream
  propagation of Rossby Wave activity in the North and South Pacific
  basins (e.g., Renwick and Revell, 1999; Wang et al. 2011, and
  references therein). The Bering Sea region is close to one
  teleconnective centers in the PNA pattern, which should make these
  rules useful indicators of weather downstream.

Refer to the Case Studies in study II.


Source (Study I): http://www.nws.noaa.gov/ost/climate/STIP/39CDPW/39cdpw-JRenken.pdf

Source (Study II): http://www.nws.noaa.gov/ost/climate/STIP/40CDPW/40cdpw-JRenken.pdf