>push vs pull
Situational. Pulling gives better penetration and higher buildup, generally with a rougher appearance. Pushing gives a flatter, smoother bead, but it can fail to fuse well, particularly on thicker base material without preheat. Push vs. pull, angle, and weave patterns also affect how the puddle acts in out-of-position welds. With skill, proper fitup, and appropriate machine settings, either can work for most tasks*. As an aside, robot welders typically run at about 90 degrees, which has characteristics in between pushing and pulling.
*Spray-arc aluminum should be pushed to assure sufficient shielding gas coverage.
>"specialize" in stick welding in general
Depends on local demand, or how far you're willing to travel. Like you said, stick is used for outdoor work, so there's demand for stick welders on projects that need to be welded in the field, like pipelines, mining equipment, and whatnot. Outside of actual professional specialization, skill with stick is good to have in your toolbox because it can handle dirt well, operate outside / without gas tanks, and specialty rods are available in small amounts for odd jobs.