Who really pays transaction taxes like VAT or sales tax? In Europe, shop prices are typically displayed including VAT. This gives the impression that the seller is paying the VAT. They have to offer a competitive price, then a percentage of that is taken out as tax. But, if VAT didn't exist, sellers could offer a lower price. So in another sense, buyers pay the tax.

In the US, sales tax is typically added at the till. This makes it feel like the buyer is paying the tax. But ultimately, VAT and sales tax are the same (at the point of final sale) - does it really make a difference how the prices are displayed?

The same issue exists with income tax, stamp duty, and presumably many other taxes. It's quite integral to Western life that income tax is paid by the employee - you hear people say "I pay my taxes, I'm part of this society". But the need to pay income tax means that employers need to pay higher wages, so the employer is bearing the cost as well.

Does it make a difference what happens when the rate changes? For example, if income tax rises, typically salaries do not rise to match. But if sales tax rises, the the amount added at the till immediately rises too.
The issue you are asking about is tax incidence. This Wikipedia entry provides a good introduction. In short: it depends on how easily supply and demand respond to price changes relatively to each other, the so called price elasticity of demand and supply.

According to standard theory, it doesn't matter who pays (in the legal/fiscal sense) the taxes, as the burden of the taxes depend on economic circumstances. If demand falls rapidly with higher prices while supply does not, sellers/producers will lower their price (before tax), and therefore carry a relatively large part of a tax.

Of course, there may be all kinds of administrative costs and other 'frictions', that make it more or less efficient to have one group or another pay taxes. Also, there is evidence that people do change their behavior according to how the taxes are paid; see for example this paper, or this document.