Players wear helmets and pads for protection. (direct objects, predicate nouns)

Question

Helmets and pads for protection.(direct object)

However, injuries can sometimes be serious or lasting. (indirect objects, predicate
adjectives)
Sometimes be serious or lasting.(predicate adjective)

Drills give the offense and the defense experience. (indirect objects, predicate nouns)
The offense and the defense experience.(Predicate Noun)

Answer 1

http://www.myenglishpages.com/site_php_files/grammar-lesson-direct-indirect-object.php

You are correct, helmets and pads is a noun phrase, the direct object. "for protection" is a prepositional phrase, not part of the noun/direct object It modifies the verb "wear".

There is no indirect object in the second sentence. "serious and lasting" is the predicate adjective phrase.

Experience is the direct object of the verb give. "the offense and the defense" are indirect objects telling to whom the experience is given.

Read about indirect and direct objects in the link above.