"Free" transactions are free because they're not immediate. The broker buys the share themselves and sells it to you at markup... ie: there is still a transaction fee, you just have no idea what it is.

Day traders use platforms that are optimized for speed and minimal fees, and that don't charge fees based on lot size.

Do you know the difference between a limit order and a market order?

Turns out most day traders are eaten alive. There's one study a few years ago that looked at Brazilian day traders and found 97% of traders that traded for more than 300 days were unprofitable [1]. I imagine this is due to a combination of factors which include 1) no real edge against the market and 2) fees. Of course unclear if their results generalize to other equity markets, but I think this is some evidence that the average day trader will have a difficult time beating the more sophisticated market participants over a large sample.

[1] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3423101

Probably not, they aren't allowed to do that and don't.

You can only get an adverse price on a market order