Quads will be internally split into two triangles, so the drawing speed should be identical. Be careful because in some situations the automatic conversion can yield unexpected results (particularly with texture coord interpolation) so you might prefer to do this step manually.

Yup, strips and fans let you specify geometry without sending as many points. Graphics cards also have a post-transform cache which reuses vertices rather than transforming them each time which saves calculations. Since the vertices are shared that means the texture coords are shared too - usually this isn't a problem as they represent a continuous surface. If you want different texture coords you'll have to add extra vertices or split things up into separate strips. You can't change texture in the middle of a strip or fan either, so you have to split them up then too. Alternatively you can pack multiple textures into one bigger texture (like a sprite sheet or texture atlas) so you don't have to break up your strips.

Fans are pretty much useless since most geometry isn't fan-like. Strips are used much less now - it's usually preferable to use index geometry which gives you the same speed advantages without the awkward step of trying to strip-ify your model data.

Changing texture every quad is going to kill your performance if you're not careful. You should look into using a texture atlas or sprite sheet so you can reduce the number of times you have to switch texture and so draw more stuff at once.