### Solution:

#### Part 1: Cost of One Desk Lamp and One Flashlight

Let's denote:
- The cost of purchasing one flashlight as $x$ dollars.
- Consequently, the cost of purchasing one desk lamp is $x + 20$ dollars, since a desk lamp costs $20 more than a flashlight.

Given that the number of desk lamps purchased is half the number of flashlights purchased, and the total spent on desk lamps is $400$ dollars and on flashlights is $160$ dollars, we can set up the following equation to represent the relationship between the cost and the number of items purchased:

$$\frac{400}{x + 20} = \frac{1}{2} \cdot \frac{160}{x}$$

Simplifying this equation:

$$\frac{400}{x + 20} = \frac{80}{x}$$

Cross-multiplying gives:

$$400x = 80(x + 20)$$

Expanding and simplifying:

$$400x = 80x + 1600$$

$$320x = 1600$$

$$x = 5$$

Thus, the cost of purchasing one flashlight is $5$ dollars, and the cost of purchasing one desk lamp, which is $x + 20$, equals $25$ dollars.

Therefore, the cost of purchasing one desk lamp is $25$ dollars, and the cost of purchasing one flashlight is $5$ dollars.

\boxed{\text{The cost of purchasing one desk lamp is } \25, \text{ and the cost of purchasing one flashlight is } \5.}

#### Part 2: Maximum Number of Desk Lamps Rongqing Company Can Purchase

Let:
- The number of desk lamps purchased be $a$.

Given the promotion and the company's needs, the number of flashlights needed is $(2a + 8) - a = a + 8$.

The total cost of purchasing desk lamps and flashlights, considering the promotion and the cost of each, should not exceed $670. Thus, we have:

$$25a + 5(a + 8) \leqslant 670$$

Simplifying:

$$25a + 5a + 40 \leqslant 670$$

$$30a \leqslant 630$$

$$a \leqslant 21$$

Therefore, Rongqing Company can purchase at most $21$ desk lamps of this brand.

$\boxed{\text{Rongqing Company can purchase at most } 21 \text{ desk lamps of this brand.}}$