ENG GRAMMER PLS

(1) the fundamental problem facing the human race

This is not a complete sentence but a noun phrase.
"the fundamental problem" itself is a noun phrase.
"facing the human race" is a participle phrase acting as adjective to modify the noun "problem". It can also be viewed as a reduction of the relative clause "that faces the human race".

To complete the whole sentence, you need to have a predicate (a verb, as a minimum requirement) placed after the above noun phrase, the subject.

e.g. The fundamental problem facing the human race [ has become more related to the issues of environmental protection ].

where, [...] is the predicate of the sentence.

(2) the profound effect global warming has on the environment

Again, this is not a complete sentence but only a noun phrase.
"the profound effect" is a noun phrase by itself.
"(that) global warming has on the environment" is a relative clause modifying the noun "effect" with the relative pronoun "that" omitted.

Note that this relative clause cannot be reduced to a participle phrase as in (1), because the modified noun is the object of the relative clause.

i.e. global warming has [the profound effect] on the environment

Again, it needs a predicate to complete the sentence.

e.g. The profound effect global warming has on the environment has increased public awareness in recent decades.