I have been careful here to cover both shares in companies and in ETFs (Exchange Traded Funds). Some information such as around corporate actions and AGMs is only applicable for company shares and not ETFs.
The shares that you own are registered to you through the broker that you bought them via but are verified by independent fund administrators and brokerage reconciliation processes. This means that there is independent verification that the broker has those shares and that they are ring fenced as being yours. The important point in this is that the broker cannot sell them for their own profit or otherwise use them for their own benefit, such as for collateral against margin etc..
1) Since the broker is keeping the shares for you they are still acting as an intermediary.
In order to prove that you own the shares and have the right to sell them you need to transfer the registration to another broker in order to sell them through that broker. This typically, but not always, involves some kind of fee and the broker that you transfer to will need to be able to hold and deal in those shares. Not all brokers have access to all markets.
2) You can sell your shares through a different broker to the one you bought them through but you will need to transfer your ownership to the other broker and that broker will need to have access to that market.
3) You will normally, depending on your broker, get an email or other message on settlement which can be around two days after your purchase. You should also be able to see them in your online account UI before settlement. You usually don't get any messages from the issuing entity for the instrument until AGM time when you may get invited to the AGM if you hold enough stock. All other corporate actions should be handled for you by your broker.
It is rare that settlement does not go through on well regulated markets, such as European, Hong Kong, Japanese, and US markets but this is more common on other markets. In particular I have seen quite a lot of trades reversed on the Istanbul market (XIST) recently. That is not to say that XIST is unsafe its just that I happen to have seen a few trades reversed recently.