Will there be any conflicts doing a mongodump on a 32-bit install and a mongorestore to a 64-bit installation?
No, this will be fine, no problems - the BSON format will not be changed because of architecture
Is there any problems moving a 2.0 database dump to a 2.2.x installation?
No problems here either, in fact you can just copy the data files directly if you wish (as long as the source DB is shut down at the time). That will in fact be faster because you won't have to rebuild indexes.