The Tamiya paint bottles are great for airbrushing, but really really bad for painting with brushes. Any I've used are far too thin to work without a lot of layers (which is what makes them great for airbrushing). They're also remarkably smelly.
If you can get the primer on then you're painting on it, not the model, so normal citadel or vallejo acrylics should be fine.
I use the cheap craft paints for scenery and weathering vehicles. People do use them for figures, but artist's paints are generally too thick to use without thinning, so I'd stick to model paints if you can.
I'm not sure about the quality of the plastic on the figure, but I've used Valspar plastic primer with success on cheap crap (tm) from Poundland. However, it's an etching primer, so you may not want to do that to the surface.
Another thing you could try is to prime it using matt varnish. I've also used that on vehicles with success. The main thing you're trying to achieve is to get something to stick to the plastic.
I'm not sure I'd bother trying to strip the paint, unless it's blocking details. You definitely want to give it a good wash in warm soapy water before priming, but as long as your primer sticks, that will cover the original paint (unless you use a clear primer, obviously) and give you a good base to start applying the main colours.