Rewrite the first mixed number to allow for subtraction of the fractions.

Since $\frac{3}{7} < \frac{4}{7}$, we borrow 1 from the whole number 7.

$$7\frac{3}{7} = 6 + 1 + \frac{3}{7} = 6 + \frac{7}{7} + \frac{3}{7} = 6\frac{10}{7}$$

$$1 + \frac{10-4}{7} = 1 + \frac{6}{7}$$

The difference is $\boxed{\mathbf{1\frac{6}{7}}}$.

Add the whole numbers and the fractions separately.

$$5\frac{4}{11} + 1\frac{5}{11} = (5+1) + \left(\frac{4}{11} + \frac{5}{11}\right)$$

$$6 + \frac{9-3}{11} = 6 + \frac{6}{11}$$

The result is $\boxed{\mathbf{6\frac{6}{11}}}$.