Any kind of cooked meat or fish and most perishable foods in general are safe as long as they are fully cooked and refrigerated within 2 hours (although the quality will deteriorate rapidly with fish).
If you plan to eat the leftovers twice then refrigerate two individual portions. Reheating the same item multiple times raises the risk of bacterial contamination. That is really the only thing you need to concern yourself with for short-term storage.
For more information see the USDA Basics for Handling Food Safely.
The only method I have personally found to be reliable for grilling/pan-frying chicken breasts to a relatively uniform doneness is to pound them very, very thin with a mallet or rolling pin. Thin, as in scaloppine-thin, so that it cooks almost instantly in the pan.
Every other stovetop-only method is almost certainly going to produce a bland, tough cut, regardless of whether you press it down or not. As noted in my comment, my usual (lazier) method that does not involve pounding is to get a nice sear in the pan, then jam in a temperature probe and bake it in the oven until it's done (the USDA recommends 165° F, I usually don't go quite that high).
If pressing the meat actually accomplishes anything at all, it would most likely be to just squeeze out whatever tiny amount of precious juices the breast does have, and possibly give you slightly more even cooking on the exterior only; it will not help to cook the interior much faster unless, as stated above, the cut has been pounded extremely thin and flat, at which point it doesn't really matter.
P.S. Salt and olive oil is a terrible "marinade" for any cut of meat, especially a chicken breast. The salt is just going to get suspended in the oil and never reach the meat at all, and the oil itself won't have much of an effect on such a lean cut. You really need to change your marinade as well, preferably to something water-based (or at least not 100% oil).
Best Answer
The USDA NAL has this to say:
In addition, you can compare the serving size weight of the breast with skin (145 g) to the weight of the breast with meat only (118 g), each derived from 1/2 chicken breast, so the skin accounts for about 18.6% of the deboned breast and 14.9% of the bone-in breast (accounting for the earlier 20%).
All in, it looks like a bone-in breast with skin is slightly more than 65% meat.
Breast meat is generally the most expensive part of the chicken to buy, even bone-in; if economy is really a factor here than you really should consider using the whole chicken.