an investor may be prepared to pay 1.25 for a share, but you are offering to sell for 1.20 -- instead of the investor paying only 1.20, the HFT jumps in, buys for 1.20 and sells for 1.25, pocketing the difference
Nope. The exchange would match all orders until the 1.25 was filled, including your 1.20 offer _at_ 1.20 -- the investor would save 0.05 per share for the percentage that your offer covered. HFT, like the rest of the world, would only see the trade.
the stock exchanges like having large number of quotes, as they charge for trades, so the more the better
More quotes != more trades. I could flood the market with offers to buy AAPL for $1 and not a single one would ever get matched. There are markets that are considering a tax on trade-quote ratios that are too low. In the case above, I don't believe a single quote was ever hit so in a market with minimum trade-quote ratios, this behavior would be taxed out of existence.