Can you buy stock at a particular second while watching it go up and down each second?

Theoretically it is possible but practically it is very difficult because it depends on the speed you contact your broker and his reaction time.

Theres a feature called "Limit" that you can use that allows you to set the exact amount that you will only sell or buy at that exact amount. It is executed as soon as and if that amount is reached. That would be the only time a tranasaction could be executed down to the second for average joes.

No, it does take a short time to execute a trade order.  You either take the market price or you can set a specific price you will pay,  You  might get it for less but they won't buy it if it costs more.

only if you pay a lot of money for a service , otherwise you are always seconds and probably millions of dollars behind the professionals