In theory, your non-marketable limit orders show up in the order book at whatever exchange they got routed to.

If you have an unfilled limit order which is resting on the book at a stock exchange, anyone with L2 data (available for single-digit dollars per month from InteractiveBrokers, among others) will be able to see it, though it won't be linked to you.

This is how order books work. If you don't want counterparties to see the fact that you want to buy or sell stock, your options include a) not telling a stock exchange to advertise your willingness to buy or sell stock by putting in limit orders, b) using a dark pool, or c) become more sophisticated with respect to your execution strategy, such as e.g. using repeated Immediate-or-Cancel orders, etc.

Yes, limit orders aren't local but sent to the exchanges like market orders except with an advertised price that they will accept.

This is called Level II data and goes beyond the simple bid, ask spread of Level I. The feeds are cheap and usually free if you have enough money in your account at any traditional brokerage.