The centre of Phuket town is the area you've circled 1. Your area 2 is Patong, which is primarily a tourist resort area - i.e. mostly bars, restaurants etc.
Which, if either, is the 'right' place depends on what you're after - e.g. If you're after loud music, parties etc then you're better off going to Patong, for a quiet beach go to one of the smaller ...
Simplest- the zoo
Phuket Zoo seems to have them based on this photo I found on the internet, which is titled: "Picture of Nycticebus bengalensis / Northern slow loris at Phuket Zoo":
Plain link attribution: https://zooinstitutes.com/animals/northern-slow-loris-phuket-zoo-14525.html
I would like to add that I don't recommend Phuket Zoo. There are better ...
I'd take it to the Turkish Airline desk and check it in there. Since all your flights have the same PNR the airlines should route it appropriately.
If in doubt you can always ask for confirmation at the Turkish Airlines desk.
I have been stranded all over South-East Asia on different magnificent islands due to bad weather or broken down ferries. Thus if it is an important flight, e.g international flight then I would never advise anyone to travel with a ferry and fly on the same day.
The time you are going for koh lipe the island will be fairly quiet, making the ferries even ...
I've looked into this on several occasions.
Based on my research there was no simple way to travel between the two islands.
The ferry takes too long and is not a direct service. (I also read on other forums that the journey can get quite unpleasant)
Road takes way too long.
Air was the best option but meant laying over for a couple of hours, and via KL ...