Determine the ratio of contributions.

Sarah's contribution: Shs 4,000,000
Moses's contribution: Shs 6,000,000
The ratio of Sarah's contribution to Moses's contribution is:

The total number of ratio parts is $2 + 3 = 5$.

Total profit made: Shs 1,500,000
Moses's share of the profit is $\frac{3}{5}$ of the total profit.

Since they re-invested the entire profit, Moses re-invested his share.

The amount of profit Moses re-invested in the business is $\boxed{\text{Shs 900,000}}$.

Initial number of birds: 2,000
New number of birds: 2,500

The percentage increase in the number of birds was $\boxed{25\%}$.

Feeds are bought every 7 days.
Books are balanced every 15 days.

Since 7 and 15 are coprime (they have no common factors other than 1), their LCM is their product.

This means both events will coincide every 105 days.

Starting date: March 1, 2024.
Number of days in March 2024 (a leap year, but March is unaffected): 31 days.
Days remaining in March after March 1: $31 - 1 = 30$ days.
Remaining days to add: $105 - 30 = 75$ days.

Number of days in April: 30 days.
Remaining days to add: $75 - 30 = 45$ days.

Number of days in May: 31 days.
Remaining days to add: $45 - 31 = 14$ days.

The remaining 14 days fall in June.
So, the date will be June 14, 2024.

They would buy the feeds and also balance their books of accounts on the same day on $\boxed{\text{June 14, 2024}}$.