To solve this problem using Gurobi, we need to define variables for each person's hours worked and then set up constraints based on the given conditions. We will also define two types of objective functions: one for minimizing the total combined paperwork competence rating and another for maximizing it.

Here is a Python code snippet that models the problem:

```python
from gurobipy import *

# Create a model
m = Model("Work Hours")

# Define variables
Ringo = m.addVar(name="Ringo", lb=0, ub=GRB.INFINITY)
Mary = m.addVar(name="Mary", lb=0, ub=GRB.INFINITY)
Jean = m.addVar(name="Jean", lb=0, ub=GRB.INFINITY)
Dale = m.addVar(name="Dale", lb=0, ub=GRB.INFINITY)
John = m.addVar(name="John", lb=0, ub=GRB.INFINITY)
Bill = m.addVar(name="Bill", lb=0, ub=GRB.INFINITY)

# Define constraints
m.addConstr(Ringo + Mary >= 25, name="Ringo_Mary")
m.addConstr(Jean + Dale >= 30, name="Jean_Dale")
m.addConstr(John + Bill >= 20, name="John_Bill")

# Add more constraints based on the problem description
m.addConstr(Ringo + Mary + Jean <= 69, name="Total_Rating_1")
m.addConstr(Jean + Bill <= 33, name="Jean_Bill")
m.addConstr(Jean + Dale <= 77, name="Jean_Dale")
m.addConstr(Mary + Dale <= 88, name="Mary_Dale")
m.addConstr(Dale + Bill <= 111, name="Dale_Bill")

# Add constraints for organization scores
m.addConstr(Ringo + Mary + Jean + Dale <= 142, name="Org_Score_1")
m.addConstr(Mary + Jean + John <= 44, name="Org_Score_2")
m.addConstr(Ringo + John + Bill <= 56, name="Org_Score_3")

# Add constraints for paperwork competence ratings
m.addConstr(Ringo + Mary + Dale <= 119, name="Paper_Rating_1")
m.addConstr(Ringo + Mary + Jean <= 116, name="Paper_Rating_2")
m.addConstr(Mary + Dale + John <= 67, name="Paper_Rating_3")

# Objective function
m.setObjective(Ringo + Mary + Jean + Dale + John + Bill, GRB.MINIMIZE)

# Solve the model
m.optimize()

# Print solution
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print("Obj:", m.objVal)
```

This code defines variables for each person's hours worked and sets up constraints based on the given conditions. It also defines an objective function to minimize the total combined paperwork competence rating.

Note: This is a simplified model that assumes all variables are non-negative and does not account for any specific relationships between the variables beyond what is specified in the problem description. You may need to modify the code to better fit your specific needs.