For Fame, just keep completing quests for people in the various towns and areas. Working as a waitress in the first town or dancing in the ice town will boost your fame a few points each time you work. Also, speak to the Guild Manager in the guild hall, she will occasionally have quests for you to complete.
I don't know which dungeon the 'deep layer dungeon' is, so I cannot help with that one.
If you are getting beat up badly by monsters in any dungeon, you are probably under leveled or under geared. So, spend some time grinding for experience and money to upgrade your party's equipment and then try that particular dungeon again.
For the horse quest, you need to build the Chapel/Magic building in your base. When you do, there will be a few books that you can read for certain quests. The horse quest is one of those quests, reading the book will allow the MC to learn that the horses are horny and that she needs to spend some time servicing their cocks to calm them down.