Hat On The Nose

There are two reasons why not to look at the face or hands of the כהנים during נשיאת כפיים.

1) היסח הדעת  (distraction) Both, the כהנים & עם must concentrate on the ברכת כהנים and not get distracted.

2) זכר למקדש  In the Bais Hamikdash, they did  ברכת כהנים with the שם המפורש. Since the שכינה hovered over the Kohanim

during נשיאת כפיים, it blinded the eyes of those who gazed at them. (only when it is done with the שם המפורש ) 

Since today staring at the Kohanim won't blind you, a short glance would be OK but because of  זכר למקדש the Minhag is to refrain even from a short glance  (משנה ברורה)

There are two Minhagim:
1) The Kohen covers only his face with the Tallis and keeps his hands exposed.

 2) He covers both his face & his hands under the Tallis. (רמ"א)

In those places where the Kohanim exposed their hands, the Minhag was also for the עם to cover their faces with their Tallis. (דרכי משה)

Today, many Yidden cover their face with the Tallis even at places where the כהנים do cover their hands. this is done because of מנהג אבותיהן even though it's not really necessary. 

Placing the hat on the nose during ברכת כהנים is definitely nonsensical.