There is no standard way to map the SOI onto El Nino or La Nina conditions, so different operational centers and research studies will use different thresholds. The Australian Bureau of Meteorology (BOM) uses thresholds of -8/+7, whereas NOAA ENSO forecasters do consider SOI and other atmospheric indicators but do not use any explicit SOI threshold for declaring El Nino/La Nina conditions. The SOI thresholds are likely to vary considerably in the research literature. It's important to consider that not all studies use the same units in their SOI time series. For example, BOM uses an SOI that multiplies the standardized pressure differences by 10. So any studies that use the -1/+1 SOI thresholds are likely not multiplying the index by 10, and so those -1/+1 thresholds really correspond to -10/+10 if we use the formula used by BOM. 

Regarding your other question, I am not sure of the source of your confusion, but I can give a brief description of the indexes. The Nino 3.4 index is the SST anomaly averaged in the Nino 3.4 region, as discussed in this post. This is the index that is directly used for monitoring and forecasting ENSO. The AO is the Arctic Oscillation, and it refers to an atmospheric circulation pattern primarily over the Arctic and North Atlantic and that is related to the polar vortex. The AO has important impacts throughout the hemisphere, primarily in boreal winter, and we discussed it in this post. The EAWR refers to the East Atlantic/Western Russia pattern, and this is another atmospheric pattern like the AO but that gets a lot less attention. There is a brief description of it on this CPC website.  

If you have additional questions, let us know.