Lack or abundance of shares available to short. What does it mean for share price in short term?

This is correct except for one detail. The price already has gone down or has risen. It's not a future effect, it's a past effect.

If something happened that made people expect the price to rise in the future, then people would immediately buy that stock to benefit from the future rise. This pushes the price up until you can no longer reasonably expect any particular future rise.

So you're right, but the price change has already happened. It cannot be the case that publicly available information about a liquid stock can make people reasonably expect the price to significantly increase or decrease in the future because people with that expectation would already buy or sell the stock (or adjust their prices for doing so) to account for the future expectation.

Having no shares available to short means they have already been borrowed and sold. The shares could be accurately valued already, or an expectation of worse news with unknown impact impairing the finances of a company. Always remember that the point of a company is to return value to shareholders, if the amount possible to return seems less likely, then you shouldn't put as much capital in the company - nobody should - so the share price should be lower.

But good news or a price rally could be amplified by shorts getting scared of their own increasing debt and forced to buy the stock, in a short squeeze.

So you can't necessarily tell about the future price action, but you can tell about what some investors will react to.

Another thing to understand is that short sellers have to borrow shares from investors. Investors have the option of allowing their shares to be borrowable and earning a little interest from that. Investors can turn that on and off. Large insiders or any large investor with a lot of common stock can cause short squeezes in this way, or simply make shares hard to borrow or very easy to borrow.