Travelling in EU?

1. Brexit aside, there's no limits on the duration for a British citizen to stay in any European Union country or any non-EU Schengen country.
To clarify, this additionally applies to the non-Schengen EU countries of Bulgaria, Croatia, Cyprus, Ireland, Romania as well as the non-EU Schengen countries of Iceland, Liechtenstein, Norway and Switzerland.

2. Travelling between Schengen countries without border controls applies to ALL citizens regardless of nationality. However, please be aware that some Schengen countries have imposed internal ID checks due to the recent European migrant crisis (this won't affect the validity of your stay, but be sure to have your passport on hand).

3. HKSAR and BNO passports only enjoy 90 days visa-free access to the Schengen Area (all Schengen countries counted as a whole), as well as non-Schengen EU countries (duration of stay separately counted between each country in this case).
Most European airports have an additional EU/EEA citizen queue which circumvents the long waiting times, and the process is much more simple as EU/EEA passports are merely checked, but not stamped, when travelling within the EU, whereas non-EU/EEA passports are stamped. (at least this was the case when I used my BC passport to travel to Norway and France)
Overall, using the BC passport would be relatively hassle-free.

1. As of this moment, a British Citizen Passport can enjoy freedom of movement (permanent residence) within any EU member states and Switzerland.

2. See above. The Schengen Agreement (the treaty creating Schengen Area) does not apply to you as you can live within any EU member states and Switzerland permanently.

3. Absolutely not.