Then, a house somewhere caches fire and the whole city burns down, because no one was willing to pay for a single fire truck.

Yes tax is often to used to pay for public goods, though fire isn't a perfect example because in some parts of the world actually fire protection is considered a product that you buy from the government and you actually can opt out of it. In that case if your home catches fire the fire fighters will turn up and do only as much work as needed to stop it spreading, but they won't bother trying to save your home:

http://www.huffingtonpost.com/2010/10/04/firefighters-watch-...

That story is pretty brutal, especially if they really did forget to pay the fee and not merely decide to opt out, though it's hard to know after the fact of course. But it is possible and it does happen.

Still, there are lots of other examples of public goods funded via taxation and vaccination is a good one.

One point of terminology confusion here: sales tax and VAT are not the same thing. Sales taxes usually only apply when goods are sold to a consumer at retail. VATs apply at all parts of the supply chain and have rules in place to charge tax only on the value delta. America uses sales tax, I think most other places (and certainly Europe) use VAT. It's very confusing to claim they are the same thing. Additionally sales taxes are not normally described as progressive - it's the opposite, they are considered regressive:

http://en.wikipedia.org/wiki/Sales_tax

(see the "effects" section)

... because the rate is the same no matter what your income is. Rich people don't necessarily consume more: they can choose to consume the same amount as poorer people and often do just because many goods are pretty uniform. I.e. rich people tend to use the same smartphones as middle or even lower income people. They don't use smartphones that cost 10,000x as much even if that's how much more they earn.

I'm not saying there should be no double tax treaties - there should be, of course, but companies should not be able pay less tax on the profits generated within one country, just because taxes in the other country are lower.

Companies don't generate profits within a country, they generate sales, and they already pay the local rates of sales tax or VAT. Attempting to bodge this arrangement is why politicians are always screaming about how unfair profit-shifting is, but it's fundamental - how do you define where a profit is made? Deciding where a sale is made is not too hard because you can identify the location of the person who bought the good or service, or where the business is headquartered, but when you add up all the profits from these sales which country gets to help themselves? Consider Google. Is their profit made in California because that's where they were founded and where the brains of the search engine are made? Or Ireland, because that's where the sales people were made? Or Switzerland because there's an office there that contributed a nifty feature that increased profits? How much does each country contribute towards a companies profit? Trying to figure this out is a mugs game, it's entirely arbitrary, which is why companies can game it so easily by having subsidiaries license IP from each other and so on - who can really say what arrangement is correct?

In theory, in the EU, it's simple: profit is booked where the company is headquartered. Same thing for the USA, hence all the companies registered in Delaware, a notorious tax haven. Sometimes you can't arrange things in this way because you need / want to operate outside that trading bloc, this is when you start getting subsidiaries that pay their parent companies huge licensing costs for IP or the brand, but really it's just about moving profits to where the company wants them to be.