What % of the average daily volume of a stock can be purchased without affecting the price of the stock significantly?

There's no way to definitively answer the question other than to say that less liquid stocks are more apt to move in price than more liquid stocks. Each day, it depends on the buy/sell dispersion in the order book. Given that, let's go with your numbers and make up some more.

If there are 20,000 shares offered for sale from $2.00 to $2.05 (ask price) and no further sellers appear, it will take the buying of 20,000 shares to move price up 5 cents.

OTOH, suppose that there are 20,100 shares offered for sale at $2.00 and buyers take out 20,000 shares at $2.00 then price will remain at $2.00

The first metric I would look at would be the average daily volume. I would only look to trade stocks with an average daily volume of at least 10x the volume I was looking to trade.

For example, if you were looking to trade 10,000 shares, you would only look for shares with an average daily volume of at least 100,000.

As a second metric I would also only trade stocks with a tight spread. Using these 2 metrics you shouldn't move the market more than a couple of cents with your trades.

You are effectively asking how many shares can be traded at or near the close of market without affecting price dramatically. The answer is on average about 3-4% of the volume near close. This varies greatly. It will be much less than 0.5% of the daily volume.

Another factor is the volatility (vol) of the instrument for the day compared to its average vol. The more abnormal the volatility, the larger volumes the close will do.