I'll do the first two problems to get you started
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Problem 1
Mary and Jenny are the two subjects of this sentence. They are the compound subject when we group them together as one. Subjects are nouns since they are a person place or thing. In this case, we're talking about two persons.
We group them because they are both performing the same action (verb) of running. Specifically they both ran in the rain.
Rain is also a noun, but it's not the subject of the sentence. The rain isn't performing an action, at least in the sense the sentence is stating. We consider this the indirect object.
The word "and" is a conjunction. It connects the two subjects to form the compound subject overall.
The word "in" is a preposition which describes how a person or object is located in relation to another. It also helps set up context.
The word "the" is an article which comes before a noun.
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Problem 2
"My" is a possessive pronoun. Other examples of possessive pronouns are: hers, his, theirs. They are pronouns that describe relationships or ownership. So in this case, the relationship here is that it's your uncle.
The subject of the sentence is the uncle. The verb or action he takes is "cook". The verb is in the future tense due to the "will cook"
The direct objects here are lunch and dinner since the verb cook connects to these objects. In other words, lunch and dinner are nouns (they are things to describe certain eating events).
The word "or" is considered a conjunction since it connects two nouns or phrases.