does not sound like it's the tooth. it sounds like the crown. it could be that the crown has a margin (edge) that is open (not flush up against the tooth) or too thick (which can irritate the gum tissue), or poorly contoured (can also irritate the gum tissue) or it could be that the edge of the crown extends too far under the gum line, which can also cause gum irritation. in any of the first 3 scenarios, the crown should be redone, as these are correctable scenarios with a proper fitting/properly contoured crown. In the 4th scenartio a periodontist may be needed to reposition the gum tissue around the tooth before the crown is remade. think 2-3 weeks is a long enough time for things to settle down. if it's been longer than that, i would consult with the dentist who made it, and if you are not satisfied, i would see a periodontist (gum specialist) to evaluate the problem. remember, it is time consuming and costly for a dentist to redo a crown, but that should not keep you from getting it done right and getting answers that satisfy you. good luck
The questions you propose are very difficult to answer with out seeing you. The soreness from the injections, if there is any, does not last a long time. If the discomfort does remain or gets worse then a visit to the dentist is in order.