Holy cow... People put a lot of nonsense down for such simple questions.
First, prepositions can act as adverbs, and prepositional phrases can act as adjectives and adverbs. As with any other adjective or adverb, you can remove them if they are not essential for understanding.
"I am going to go wash my red corvette." 'Red' is an adjective and not essential.
"I am going to go wash my corvette."
If I said, "I am going to go wash my red corvette with soap and water." My prepositional phrase "with soap and water" is acting adverbially to modify the verb "wash"; it is also non-essential and I can get rid of it.
In your first example:
"I bought chopsticks for eating rice with." "for eating rice" is a prepositional phrase acting adjectivally to describe "chopsticks". Most all of us know what chopsticks are for, which is why native speakers are telling you to get rid of it. They know it "sounds bad", but no one seems willing to tell you why it sounds bad (not accurately, at least).
"with" is no longer acting as a preposition, but as an adverb to modify the verb "eating". It is non-essential and can be removed.
The next example, "I bought a pair of chopsticks to eat rice with." is NOT using a prepositional phrase as the modifier; "to eat rice with" is an infinitive phrase.
An infinitive phrase can act as a noun, an adjective, or an adverb.
That means an infinitive can be removed when it is an adjective or adverb, but not when it is a noun since it will be performing an essential function.
For example, in "I love to eat." 'to eat' is an infinitive forming the direct object of the transitive verb "love". If we take it out, we will deprive our sentence of meaning... "I love..."
The reason this infinitive "sound better" to all these native speakers is because it is not describing the chopsticks, it is acting adverbially to say why you bought the chopsticks. This is not essential to the meaning unless you need the reader to understand that the chopsticks are for rice and not noodles. If that is essential to the understanding your purchase, you would need it:
"My old chopsticks are for eating noodles."
"I bought these chopsticks to eat rice with."
If you just said "I bought these chopsticks.", it would deprive the sentence of its essential meaning.
Same application to your last sentence:
"I bought a house to live in." uses an infinitive phrase "to live in" adverbially to establish why you bought a house, or it could be used adjectivally to describe the house, depending on context. You only need to keep this infinitive if it is essential to meaning.
"George bought a house on this street to lease out, but 'I bought a house [to live in]." "to live in" is essential to meaning and cannot be omitted.
"in" is no longer acting as a preposition, but as an adverb to describe where you "lived"--You live in the house, not on the house.