John B,
You are very fortunate to have so much information about your dad's service right at your fingertips. Very fortunate.
The two pins on the uniform lapels are called Distinctive Unit Insignia (DUI), and these DUI are for the 35th Signal Company.
As others have said, the patch above the right breast pocket is the Honorable Service or Discharge patch -- it was nicknamed "the ruptured duck", and was given to military personnel when they mustered out of service. (In the medal presentation box that you have, you will also see it came as a lapel pin.) Honorably discharged personnel received this to wear on their uniforms even though they were no longer in service because they most often did not have civilian clothes immediately available to them. Wearing the patch on the uniform would have provided some benefits as they travelled home, as would wearing the lapel pin on their civilian clothes.
As others have said, the ribbons over the left pocket are (as pictured) American Campaign; Europe-Africa-Middle East Campaign, with 5 bronze campaign stars; American Defense Sevice; and Army Good Conduct. The medals in your presentation box are the same, except it also has the Occupation Medal.
The rectangular red pin at the bottom of the case is the Meritorious Unit Commendation. Your dad's uniform shows the earlier, embroidered version on the lower right sleeve, but at some point the Army changed that over to the red pin; it would be worn centered over the right breast pocket (not both types at once, however.)
The European Campaign medal in the box has 1 silver campaign star; this is the same as having the 5 bronze campaign stars as shown on the medal on the uniform. The campaigns that the 35th Infantry Division were credited with were: Normandy, Northern France, Alsace-Ardennes, Rhineland, and Central Europe.
If you wanted to complete the ribbon rack on the uniform, you only need to add the Occupation medal. If you want the rack to be really accurate, they just need to be placed differently, as they are "out of order" -- the military uses an order of precedence for wearing the decorations. I believe the proper placement would be (from the viewer's perspective): top row, left to right -- Good Conduct, American Service (yellow), and American Campaign (blue); bottom row, left to right -- European Campaign, WWII VIctory, and then the Occupation medal.
I had to laugh when I read that your dad spent the was "driving a jeep lost, reading maps that made no sense, and trying not to cross over enemy lines..." My dad was a radio operator in a anti-aircraft battalion, and he said practically the same thing!
You are fortunate to have these items as well as so much of your dad's information and memories. Take good care of them. I know that I can't measure up to any of these men & women who served, but I can be a steward and pass their memories on.
And hey -- hang around Whiteblaze and go for a hike!