It's similar, but not exactly the same. Strava uses a different formula for Weights Power (Normalized Power in Training Peaks terminology). Above-threshold efforts are more heavily weighted in the TrainingPeaks/Coggan formula than in the Strava formula. Since normalized power is used to calculate TSS and all the other metrics, that means the Strava numbers will be a little different, especially if you're doing lots of high-intensity stuff.
The other big limitation with Strava is historical data. In Strava everything is based on your current FTP setting. So if your FTP is changing, the historical data becomes inaccurate.
Yes and no.
From what I have read strava's weighted average works the same as NP; with one exception. Strava uses a 25s rolling average before subjecting to to quartic power/root, whereas TrainingPeaks/NP uses a 30s rolling average.
Obviously, if we used a 1s rolling average, then AP = NP would always be the case. The larger the rolling average, the more higher watts get stressed.
In other words, if you had an FTP of 300 and rode for 15' at 330w and that was your entire ride NP and WAP would be identical. Strictly speaking it's not that efforts higher than threshold are more heavily weighted (because a riders threshold will not effect the NP of the ride), it's that all efforts are more heavily weighted under NP, whether they are 200w, 600w, or 1200w, but because of the effect of raising data to the 4th power, summing it, and then taking the quartic root, the higher wattage efforts are more punished the smaller the window for rolling average is.