How many tins were packed in the first layer?

The length of Box A is $56 \text{ cm}$. The diameter of Tin B is $7 \text{ cm}$.

$$\text{Number of tins along length} = \frac{\text{Length of Box A}}{\text{Diameter of Tin B}} = \frac{56 \text{ cm}}{7 \text{ cm}} = 8 \text{ tins}$$

The width of Box A is $35 \text{ cm}$. The diameter of Tin B is $7 \text{ cm}$.

$$\text{Number of tins along width} = \frac{\text{Width of Box A}}{\text{Diameter of Tin B}} = \frac{35 \text{ cm}}{7 \text{ cm}} = 5 \text{ tins}$$

$$\text{Tins in first layer} = \text{Tins along length} \times \text{Tins along width} = 8 \times 5 = 40 \text{ tins}$$

The number of tins packed in the first layer is $\boxed{40 \text{ tins}}$.

The height of Box A is $60 \text{ cm}$. The height of Tin B is $10 \text{ cm}$.

$$\text{Number of layers} = \frac{\text{Height of Box A}}{\text{Height of Tin B}} = \frac{60 \text{ cm}}{10 \text{ cm}} = 6 \text{ layers}$$

$$\text{Total tins} = \text{Tins in first layer} \times \text{Number of layers} = 40 \times 6 = 240 \text{ tins}$$

The total number of tins that were packed in the box is $\boxed{240 \text{ tins}}$.