### Solution:

#### Part 1: Finding the Purchase Price per Pair

Let's denote:
- $x$ as the purchase price per pair of speed skating shoes in dollars.
- $y$ as the purchase price per pair of figure skating shoes in dollars.

From the given information, we can set up the following system of equations:

1. For 30 pairs of speed skating shoes and 20 pairs of figure skating shoes, the total cost is $8500:
$$30x + 20y = 8500$$

2. For 40 pairs of speed skating shoes and 10 pairs of figure skating shoes, the total cost is $8000:
$$40x + 10y = 8000$$

To simplify, we can divide the first equation by 10 and the second equation by 10, resulting in:

$$\begin{cases} 3x + 2y = 850 \\ 4x + y = 800 \end{cases}$$

Solving this system of equations, we find:

- Multiplying the second equation by 2 to eliminate $y$:
$$4x + y = 800 \Rightarrow 8x + 2y = 1600$$

- Subtracting the first equation from this result:
$$8x + 2y - (3x + 2y) = 1600 - 850$$
$$5x = 750$$
$$x = 150$$

- Substituting $x = 150$ into one of the original simplified equations:
$$3(150) + 2y = 850$$
$$450 + 2y = 850$$
$$2y = 400$$
$$y = 200$$

Therefore, the purchase price per pair of speed skating shoes is $\boxed{150}$ dollars, and the purchase price per pair of figure skating shoes is $\boxed{200}$ dollars.

#### Part 2: Minimum Number of Pairs of Speed Skating Shoes

Let $m$ represent the number of pairs of speed skating shoes purchased. Then, the number of pairs of figure skating shoes purchased is $(50 - m)$.

Given the prices found in part 1, the total cost for purchasing $m$ pairs of speed skating shoes and $(50 - m)$ pairs of figure skating shoes should not exceed $8900:

$$150m + 200(50 - m) \leq 8900$$

Simplifying this inequality:

$$150m + 10000 - 200m \leq 8900$$
$$-50m \leq -1100$$
$$m \geq 22$$

Therefore, the minimum number of pairs of speed skating shoes the school should purchase is $\boxed{22}$.