To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's denote the number of couches as \(x_1\) and the number of beds as \(x_2\).

The objective is to maximize profit. The profit per couch sold is $200, and the profit per bed sold is $400. Therefore, the objective function can be represented as:
\[ \text{Maximize:} \quad 200x_1 + 400x_2 \]

The constraints are as follows:

1. **Space Constraint**: Each couch takes 15 sq ft of space, and each bed takes 20 sq ft of space. The total available space is 300 sq ft.
\[ 15x_1 + 20x_2 \leq 300 \]

2. **Minimum Beds Constraint**: At least 50% of all items in stock must be beds.
\[ x_2 \geq 0.5(x_1 + x_2) \]
Simplifying, we get:
\[ x_2 \geq 0.5x_1 + 0.5x_2 \]
\[ 0.5x_2 \geq 0.5x_1 \]
\[ x_2 \geq x_1 \]

3. **Budget Constraint**: The store wants to spend at most $8000. Each couch costs $300, and each bed costs $600.
\[ 300x_1 + 600x_2 \leq 8000 \]

4. **Non-Negativity Constraints**: The number of couches and beds cannot be negative.
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Now, let's represent this problem in the requested format:

```json
{
  'sym_variables': [('x1', 'number of couches'), ('x2', 'number of beds')],
  'objective_function': '200*x1 + 400*x2',
  'constraints': [
    '15*x1 + 20*x2 <= 300',
    'x2 >= x1',
    '300*x1 + 600*x2 <= 8000',
    'x1 >= 0',
    'x2 >= 0'
  ]
}
```

And here is the Gurobi code to solve this problem:

```python
from gurobipy import *

# Create a new model
m = Model("Furniture_Store_Profit_Maximization")

# Create variables
x1 = m.addVar(lb=0, vtype=GRB.INTEGER, name="number_of_couches")
x2 = m.addVar(lb=0, vtype=GRB.INTEGER, name="number_of_beds")

# Set the objective function
m.setObjective(200*x1 + 400*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(15*x1 + 20*x2 <= 300, "space_constraint")
m.addConstr(x2 >= x1, "minimum_beds_constraint")
m.addConstr(300*x1 + 600*x2 <= 8000, "budget_constraint")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of couches: {x1.x}")
    print(f"Number of beds: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```