Yes, stock price is determined by the last trade price.
There are always going to be people who have put in a price to buy a stock (called a bid price) and people who have put in a price to sell a stock (called an ask price).
Based on your example, if the last trade price for the stock was $1.23, then you might have the following bid prices and ask prices:
- bid: 50 shares at $1.20
- bid: 20 shares at $1.21
- bid: 30 shares at $1.22
- ask: 40 shares at $1.23
- ask: 15 shares at $1.24
- ask: 45 shares at $1.25
So if you put in a limit order to buy 100 shares at $100, you would buy the 40 shares at $1.23, the 15 shares at $1.24, and the 45 shares $1.25. The price of the stock would go up to $1.25.
Conversely, if you put in a limit order to sell 100 shares at $0.01 (I don't think any broker would allow a sell price of $0.00), you would sell 30 shares at $1.22, 20 shares at $1.21, and 50 shares at $1.20. The price of the stock would go down to $1.20.