Note that the ambient spec is as given in the notes, whatever board and copper they used for the test.
0. Do you know that you need improved thermals? Typically, if you need higher ambient temperature operation, you would. Or maybe operating at (or beyond?) ratings, if it's derated a bit to begin with.
1. Typical builds on multilayer boards have good thermal performance already, given you've used inner layer planes, which spread the heat out. Heat conducts reasonably well between top and inner layers, where the laminate is thin (a typical proto board has 5-10 mils of prepreg here).
2. Thermal vias are fine, and can also be filled with solder to approximately double the thermal conductivity; beware of solderability issues with the BGA/LGA device however. You don't want the vias thieving solder from the pads. Place them between pads if possible, and tent with soldermask.
3. Screws? Even with copper screws (which you probably can't get -- aluminum, brass and bronze, yes, but they also conduct significantly worse than pure copper), you have the problem of getting heat over to them; lateral heat spreading is not very good in PCBs. Consider using a heat spreader at least, but even better, a thermal pad or something.
4. Don't worry about soldermask, by the way. It's thin, a tiny fraction of the total thermal resistance. It hardly matters, even if you're using a thermal pad to sink heat out directly by conduction. It likely improves performance in a passive convection scenario (the slight edge coming from the higher IR emissivity of the coating).
Tim