Yup.
Yes and no. You can:
1) Calibrate to the elevation where you are, if you know it,
2) Calibrate to the barometric pressure where you are, if you know it (this is the sea-level adjusted barometric pressure reported by weather stations everywhere; no fancy calculations required. A weather station within 25 miles is plenty close for this purpose)
3) Calibrate to the current GPS elevation. In this case you are at least as well off as those who don't have barometric sensors, since GPS elevation is all that their receiver can give them. Unfortunately, GPS elevation is subject to relatively large errors, and it changes constantly as satellite geometry changes; but it is pretty good as a starting point for autocalibration (see below) if you don't know your current elevation or barometric pressure, or
4) Calibrate to nothing, and eventually (30 minutes to an hour in typical situations) autocalibration (if you have it turned on) will still get you pretty close to a "real" elevation.
That isn't a problem if you turn on autocalibration. Autocalibration compares the difference between the elevation determined by the barometric sensor with a somewhat long-term average of GPS elevation, and calibrates the barometric elevation to minimize the difference. Over a period of time (don't as me how long), the average GPS elevation is relatively accurate. By calibrating the barometric sensor to the average GPS elevation, the effects of pressure changes are compensated for. Automagically. It's very sophisticated, and it works very well. Mine corrected a deliberate miscalibration of 100 feet in about an hour. That's much faster than barometric pressure normally changes.
Amazingly, autocalibration seems to work well over the length of a day even when I turn my receiver on only occasionally to get a position fix and then turn it off again to save batteries. A fellow named Chris Malcolm once documented that the algorithm seems to have what he called a "fast catch-up mode" to handle this situation. Unfortunately, Mr. Malcolm's excellent web page on autocalibration is no longer where it once was, and I've been unable to find it elsewhere.
Of course, even the average of GPS elevation isn't perfect, so your receiver may still be off some, but it's pretty good, and definitely better than GPS elevation alone. The downside is that for periods of time that are short enough for the barometric pressure to be constant, autocalibration may actually reduce accuracy, as hogrod stated very well previously in response to the OP: