Let's solve each of these mathematical expressions step-by-step:

### 1. \( \frac{1}{3} \left( \frac{6}{5} - \frac{9}{4} \right) \)

First, find a common denominator for the fractions inside the parentheses:
\[ \frac{6}{5} = \frac{24}{20} \]
\[ \frac{9}{4} = \frac{45}{20} \]

Now, subtract the fractions:
\[ \frac{24}{20} - \frac{45}{20} = \frac{24 - 45}{20} = \frac{-21}{20} \]

Then, multiply by \(\frac{1}{3}\):
\[ \frac{1}{3} \times \frac{-21}{20} = \frac{-21}{60} = \frac{-7}{20} \]

### 2. \( -\frac{7}{5} \left( \frac{15}{14} + \frac{5}{7} \right) \)

First, find a common denominator for the fractions inside the parentheses:
\[ \frac{15}{14} = \frac{15}{14} \]
\[ \frac{5}{7} = \frac{10}{14} \]

Now, add the fractions:
\[ \frac{15}{14} + \frac{10}{14} = \frac{25}{14} \]

Then, multiply by \(-\frac{7}{5}\):
\[ -\frac{7}{5} \times \frac{25}{14} = -\frac{7 \times 25}{5 \times 14} = -\frac{175}{70} = -\frac{5}{2} \]

### 3. \( \frac{1}{5} \div \frac{3}{10} + \frac{5}{6} \)

First, perform the division:
\[ \frac{1}{5} \div \frac{3}{10} = \frac{1}{5} \times \frac{10}{3} = \frac{10}{15} = \frac{2}{3} \]

Then, add \(\frac{5}{6}\):
\[ \frac{2}{3} + \frac{5}{6} = \frac{4}{6} + \frac{5}{6} = \frac{9}{6} = \frac{3}{2} \]

### 4. \( -\frac{4}{5} \div \left( \frac{20}{9} - \frac{8}{3} \right) \)

First, find a common denominator for the fractions inside the parentheses:
\[ \frac{20}{9} = \frac{20}{9} \]
\[ \frac{8}{3} = \frac{24}{9} \]

Now, subtract the fractions:
\[ \frac{20}{9} - \frac{24}{9} = \frac{20 - 24}{9} = \frac{-4}{9} \]

Then, perform the division:
\[ -\frac{4}{5} \div \frac{-4}{9} = -\frac{4}{5} \times \frac{9}{-4} = \frac{36}{20} = \frac{9}{5} \]

### 5. \( \frac{10}{7} \div \frac{5}{14} - \frac{2}{3} \)

First, perform the division:
\[ \frac{10}{7} \div \frac{5}{14} = \frac{10}{7} \times \frac{14}{5} = \frac{140}{35} = 4 \]

Then, subtract \(\frac{2}{3}\):
\[ 4 - \frac{2}{3} = \frac{12}{3} - \frac{2}{3} = \frac{10}{3} \]

### 6. \( -\frac{3}{4} \div \left( \frac{1}{4} - \frac{5}{8} \right) \)

First, find a common denominator for the fractions inside the parentheses:
\[ \frac{1}{4} = \frac{2}{8} \]
\[ \frac{5}{8} = \frac{5}{8} \]

Now, subtract the fractions:
\[ \frac{2}{8} - \frac{5}{8} = \frac{2 - 5}{8} = \frac{-3}{8} \]

Then, perform the division:
\[ -\frac{3}{4} \div \frac{-3}{8} = -\frac{3}{4} \times \frac{8}{-3} = \frac{24}{12} = 2 \]

### 7. \( \frac{5}{26} - \frac{5}{7} \div \frac{2}{7} \)

First, perform the division:
\[ \frac{5}{7} \div \frac{2}{7} = \frac{5}{7} \times \frac{7}{2} = \frac{5 \times 7}{7 \times 2} = \frac{5}{2} \]

Then, subtract \(\frac{5}{2}\) from \(\frac{5}{26}\):
\[ \frac{5}{26} - \frac{5}{2} = \frac{5}{26} - \frac{65}{26} = \frac{5 - 65}{26} = \frac{-60}{26} = -\frac{30}{13} \]

### 8. \( \frac{3}{4} \div \left( -\frac{3}{5} \right) + \frac{1}{2} \)

First, perform the division:
\[ \frac{3}{4} \div \left( -\frac{3}{5} \right) = \frac{3}{4} \times \left( -\frac{5}{3} \right) = -\frac{15}{12} = -\frac{5}{4} \]

Then, add \(\frac{1}{2}\):
\[ -\frac{5}{4} + \frac{1}{2} = -\frac{5}{4} + \frac{2}{4} = -\frac{3}{4} \]

### 9. \( \frac{1}{3} \cdot \left( \frac{2}{15} - \frac{4}{9} \right) \div \frac{1}{9} \)

First, find a common denominator for the fractions inside the parentheses:
\[ \frac{2}{15} = \frac{6}{45} \]
\[ \frac{4}{9} = \frac{20}{45} \]

Now, subtract the fractions:
\[ \frac{6}{45} - \frac{20}{45} = \frac{6 - 20}{45} = \frac{-14}{45} \]

Then, multiply by \(\frac{1}{3}\):
\[ \frac{1}{3} \times \frac{-14}{45} = \frac{-14}{135} \]

Finally, perform the division:
\[ \frac{-14}{135} \div \frac{1}{9} = \frac{-14}{135} \times 9 = \frac{-14 \times 9}{135} = \frac{-126}{135} = -\frac{14}{15} \]

So, the results are:
1. \(-\frac{7}{20}\)
2. \(-\frac{5}{2}\)
3. \(\frac{3}{2}\)
4. \(\frac{9}{5}\)
5. \(\frac{10}{3}\)
6. \(2\)
7. \(-\frac{30}{13}\)
8. \(-\frac{3}{4}\)
9. \(-\frac{14}{15}\)