Do lava lakes exist in all volcanoes initially?
No, they do not.

Lava lakes require the lava to have low viscosity (that is, be very fluid or water like). If you look at the Wikipedia article for Lava lake, you will see that "basalt" or "phonolite" are mentioned. Those are indeed types of lavas (actually rock) that are very fluid. These are the kind of lava that end up forming lava flows.

On the other hand, there are other types of lavas that do not flow, because their viscosity is too high: rhyolite is one example. Explosive volcanoes (think Mt St Helens) are explosive (mostly) because of the high viscosity of the lava. Therefore, volcanoes erupting explosively (almost) never have lava lakes, because the lava itself is not fluid enough to make a lake.