Brakes that vibrate when engaged, worse at higher speeds indicates the rotors are warped. This can happen for many reasons, hot brakes and drive through a cold water puddle for example. If the steering wheel shakes when you brake, it is the front brake rotors with the issue. If the steering wheel does not shake it is probably the rear rotors. The mechanic that looked through the wheels and said you didn't need the rotors machined was probably saying there was no unusual wear and/or the brake pads have not scarred the rotors. The only way to determine which rotor (or rotors) is to blame is to use a dial indicator to measure the rotor face while turning the wheel.

Brakes that vibrate when engaged, worse at higher speeds indicates the rotors are warped. This can happen for many reasons, hot brakes and drive through a cold water puddle for example. If the steering wheel shakes when you brake, it is the front brake rotors with the issue. If the steering wheel does not shake it is probably the rear rotors. The mechanic that looked through the wheels and said you didn't need the rotors machined was probably saying there was no unusual wear and/or the brake pads have not scarred the rotors. The only way to determine which rotor (or rotors) is to blame is to use a dial indicator to measure the rotor face while rotating the wheel. You can have them machined, or you can replace them. Some rotors are very inexpensive and cost about what it takes to have them machined.

As for the low brake pedal, that is another issue. You may have a rusted/stuck caliper slide or, depending on the style of emergency brake, the rear calipers could have an internal emergency brake related issue. More information would be needed to go forward with that answer.

I am assuming there has not been any recent work on the brakes. A brake pad not fully inserted into the holder could also get "caught" and cause the low pedal issue.