Agricola Solo "Campaign" professions
Yes, you need to pay 1 food for your second occupation.  Note that the action space on the board reads:



  A player's first occupation is free, each additional one costs 1 food


It doesn't say anything like "first occupation played from your hand" or anything like that.


Yes, you get bonus points for any occupation played after the Home Teacher.  Per the solo rules, you choose the occupations as you play them.  There is an implied ordering, just make sure you chose the Home Teacher first!