Full disclosure - I gave my ice cream maker away after two frustrating summers. Growing up, homemade ice cream at my grandparent's farm was a weekly Sunday tradition and I was bound and determined to carry it on. I also experienced the ice issue to some extent.
Not enough fat or sugar? Very likely. My food science knowledge is fussy but a certain amount of sugar is needed to aid in the freezing process. Try making a simple syrup - add sugar to boiling water and stir very well until dissolved - as a sweetner. This will blend well into your base. Simple syrup is used in many old-timey cocktails as well as a sweetener for ice tea and lemonade. I used to bartend and it definately produces a better result. The sweetness is intergrated into the liquid in a different way then just adding granulated sugar. As I got deeper into my research, I ran into many recipes that called for corn syrup as an ingredient that aids in creating a smooth texture for homemade ice cream. I didn't have corn syrup on hand so I used simple syrup.
Fats in the base will make it much creamier. My best results were with custard bases but those require yolks. Also, I had better results using heavy cream instead of milk.
There is a famous ice cream lady whose name I can't remember (Jenny maybe?) that released a book a while back all about making ice cream (she owns a chain of gourmet shops) and she uses cream cheese in many of her bases.
Concerning the uneven freezing - Are you pre-chilling your base? If not, try chillling it in the fridge for 24 hours prior to when you put it in the maker.
Any kind of home-use maker will only get the ice cream so firm. Like Cynthia mention, I had attempts where my base went from looking good and creamy to breaking down in a way, seperating a bit. My maker called for 20 minutes of mixing and then finishing the hardening in the freezer.