When you compress the caliper pistons the brake fluid has to go somewhere, and if the master cylinder has had fluid added since the last set of pads were installed you will have too much fluid and the master cylinder will overflow.
Most shop manuals say to remove some fluid from the master cylinder to prevent this. I personally work at a higher level. I don't like brake fluid being pushed backward in the system. If done too quickly or if there is some deposits from old fluid built up, you can collapse a cup in the master cylinder so that the master will need to be overhauled or replaced. It is my personal practice to open the bleed port on the caliper when pushing the piston back. That will keep fluid from being pushed backward. If done properly you will not introduce air into the system. You should always check the fluid level when you finish a brake job. If in doubt, bleed the system. Even better, replace all of the fluid ocassionally. If not when doing a brake job, at least every two or three years.
The feeling of slightly spongier brakes may or may not be from air in the system. New pads before they are seated in usually are not as responsive as the old brakes were, requireing more pedal effort and resulting in more pedal travel.