Forests and deserts
Forests an be pretty much anywhere; there are lots of different types of forests. Having a forest and desert on the same island is pretty easy too. The island of Madiera (in the Atlantic) has a semi-arid southern coast, and a wet forested northern coast.
The key to this wet/dry divide is for the island to be in a cold current around 30 degrees north or south. Prevailing winds hit the one side of the island and deposit moisture, while the cold current reduces cloud formation over the dry side of the island. The Hadley cell providing dry descending air will complete the picture. Madiera is only 801 km$^2$, about the size of Singapore, so this effect can be prominent in a dry island. There isn't a cold current off the coast of the Eastern US, you can see here, but such an island could exist off Western Australia, Baja California, Peru, Angola, or Morocco (which is where Madiera is).
Cities can go pretty much anywhere, and are pretty straightforward. You can look up city sizes on Wikipedia to see how big a city will be for a given population.
The moist adiabatic lapse rate is the rate at which temperature drops off with altitude and is about 5C / km. If we take Madiera as an example, the capital (Funchal) has sufficient winter moisture for snow, and just needs to be colder. Since the daily mean is about 16.7C, we'd need to drop that by about 15-20 C to see regular snowfall on the top in winter. That means we are looking for mountains in the 3-4km range. This makes sense, since Mauna Loa and Mauna Kea at around 4205 m have snow on their summits a few degrees farther south on Hawaii Island.
The smallest island with a 4km mountain is that same Hawaii, at 10430 km$^2$. Going down the list, there are several that have a 3km peak while being only about 2000 km$^2$; Bioko, Siple, Tenerife, Reunion, and Ross Island. So we could expect a 2000 km$^2$ island to be able to support a snowy mountain.
What sort of lake do you want? The easiest way to do this is with a crater lake. For example Lombok has a 3700m mountain with a sizable crater lake on top of it. Lombok is about 4000 km$^2$, but any of those other islands with a comparable volcano could have a similar lake.
Your island must be at a minimum 2000 km$^2$, which is around the size of Maui, or a little more than half of Long Island. This leaves about 1/3 of the island to be covered in a city-state the size of Singapore, 1/3 to be covered in desert and forest, and the last 1/3 to be the steep slopes and summit of a mighty volcano, topped by a crater lake.