I'm working with QGIS (2.18.13) and I have a vector layer made up of many polygons (CRS: WGS84/29N) but need to change project CRS to match this in order to extract correct area data. Project CRS = WGS 84/Pseudo Mercator EPSG:3857. The base map I am using is Bing Maps Aerial (accessed through Open Layers plugin). How do I do this without distorting the polygons and keeping them in place on the map?
What really deforms and goes out of place, to establish in the project a CRS different from EPSG:3857, are the images provided by the plugin. You can use them only as a background reference, as long as your project is in EPSG:3857.
To be able to work with your layers normally, in a valid projection for your work area, you will have to turn off the Bing Aerial layer, change the CRS of the project to match that of your layer, and do not worry about the loss of spacial relationship with respect to the images of the plugin. Your polygons will be much less deformed and much more in line with their real surface if you project them on a coordinate system suitable for your work area.
If you still want to have a satellite image in the background, it would be better if you download it from the web and upload it to the project as a raster file, so that it can be reprojected to another CRS without being deformed, as the tiles served directly from the plugin.
I use Terra Incognita software for that. Select the map source, the region of interest, and the zoom level you want to download (if you abuse the zoom, the files become really heavy). Save as OziExplorer map, add the generated .map file as a raster layer in QGIS and assign the CRS EPSG:54004 (World_Mercator) to that layer.
I'm sure there are other ways. I do that because it works for me, and I never set out to investigate much more about better ways of doing it.
The raster layer will be projected without problems to the CRS that you decide for your polygons and your project. Do not forget that it is only a background reference. The service of those images is prepared just for that.