1. Either 'everyone' or 'every one' would make for a grammatical sentence, but only 'everyone' means specifically 'everybody/all the people'. 'Every one', on the other hand, could refer to 'every dog', 'every table', etc.!
2. Yes, purists will still insist on 'his/her' as the correct form, though 'their' as a generic possessive is widely tolerated in current usage.