Let's say it's communal property. In modern parlance, we might say public property. Suppose in the Great Transition from the current democracy to your libertarianism, the people can't agree as to who should own the lake. Or suppose you own one little bit of the lake, and the fishermen each own a little piece. Two possibilities for you to consider.
All property will be either owned or unowned. If it's owned then it's owned by one or more people. If it's owned by more than one person it is settled by vote.
Let's say the seller is not aware of the danger presented by the chemicals I'm using. But by the time people figure out the damage done (could be years, could be thousands of people affected), the seller has shut up shop, and moved to the next town/city/country. Of course, the danger is unknown because there's no FDA (or equivalent) to carry out the *extremely expensive* research required, and there's no law saying you can't add arbitrary stuff to the food you sell.
There could be several competing versions of the FDA. Each one with different costs and different standards. There could be super-expensive and super-safe food for paranoid people. There could be average-cost food with safety comparable to current standards. There could be low-safety standards for people that don't care or don't value safety that highly. It would ultimately be up to market forces and each person to decide what's right for them rather than "one size fits all".
Suppose the pedestrians are walking through the town square. Is that also private property in your world? Do you have to read the Terms & Conditions of the town square, and sign your acceptance, before entering it?
Yes, you would have to agree to the terms and conditions and buy a ticket signifying your acceptance much like any other private park, Disney World, etc. By the way, where would you rather meet me after dark openly holding a hundred dollar bill, in Disney World or Times Square?
What if you own a business on the town square, but reject the rules of whoever owns the road?
I would hope business owners would consider that when locating their business in the first place. Also, since road owners are trying to attract customers and having business adjacent to their roads will do that, they are going to be very reasonable if they want to make money.
Do I have to get one drivers licence for when I drive on RoadCompanyA's roads, and another license for RoadCompanyB's roads? Are there road-police checking if people are drunk-driving and checking people's cars when they drive on these private roads, one set of police for each road-owner?
That's up to each road owner to decide. Maybe you have to pay a monthly fee. Maybe you have to have an RFID chip on your bumper. Maybe you have to blow into a breathalizer before getting on the road. Maybe you have to agree to be publicly executed if you are caught driving drunk. It will be up to the road owners but since they are trying to attract customers, you can bet it won't be too restrictive or too relaxed. Whoever can provide the safest, cheapest and overall best roads will attract more customers and drive the others out of business.
I just can't see how road safety could possibly be improved in a libertarian world.
You just need to understand market forces. Currently, about 40,000 Americans die on the road each year. Does the road owner go lose money or go out of business because of that? No. So what's the incentive to improve that? Very little. However, if it were privately run, losing money would surely provoke a response.