Q. Those should be "dedicated" point to point connections, so no regular "Access Points".
A. You still have an access point. What you are thinking of using a PtP system where you have two and only two radios for each link. If you have a single AP that connects to both your neighbor and your dad, that is called a PtM link and has AP EIRP limitations becasue it's PtM and no longer PtP.
Q1. What kind of hardware do I need? Should I take a standard cheap $40 802.11n router with a standard RP-SMA plug.
A. Get a "real" system. Ubiquiti Networks, Inc.
has what you need. For your neighbor's hop, a pair of Loco M2 radios. For across town, a pair of NanoBridge M5 radios.
Q2. Apparently, this is the crucial question: which antennas should I get?
A. Many wireless systems have built in antennas. The above mentined Loco and Nanobridge have built in antennas.
Q3. The antennas have all kinds of different outputs (measured in dBi or similar). Do some of them come with an "Amplifier" that needs additional power? I find it hard to believe that a passive antenna can amplify a signal
A1. Antenna have different gains listed by dBi.
A2. Never use an extra amplifier. Seme "antennas" have the radio built in. See above for radios with built in antennas.
A3. Passive reflectors would need to be the size of an outdoor advertising billboard,
Q4. Could I mount those antennas indoors, e.g. behind a window, or would the glass already be too much of an obstacle for the connection?
A. Some glass is treated with a metallic film and would block the signal. Anything not mounted outside on a roof would have low signal levels.
Q5. How long can the antenna wire (the wire from the AP to the antenna) be at most? Is there a limit?.
A. Almost all solutions use the radio mounted right next to the antenna instead of very lossy long lengths of coax. Use CAT5 cable up to330 feet.
q6. How does a point to point wifi setup differ from a regular setup in terms of configuration? I guess you still need some kind of SSID? Or can you actually configure routers to have a single point-to-point connection - possibly even skipping TCP/IP protocol altogether and use the Wifi as if it was a hardware link.
A. All wireless setup use an AP with an SSID that a client associates to the AP's SSID. Everything is standard TCP/IP ethernet networking.