Yes. These questions are related, but the third group works differently. For the first group, all three versions are grammatically possible. “Plenty of money with which to carry out the project” is correct, but very formal. “Plenty of money to carry out the project with” is also correct. It sounds more conversational. “Plenty of money to carry out the project” is correct and is the most natural choice in most situations. They basically have the same meaning. In normal English, we do not need “with” because it is already clear that the money will be used to carry out the project. For the second group, the same rule applies. “He needs plenty of money to carry out the project” is the most natural. “He needs plenty of money to carry out the project with” is also correct, but the final “with” is not necessary. “He needs plenty of money with which to carry out the project” is grammatically correct but quite formal. The third group is different because we normally say “time to do something.” “We need some time to rest” is correct and very natural. “We need some time for which to rest” does not sound natural. “We need some time which to rest for” is not correct. “We need some time to rest for” sounds incomplete in this context. “For” would normally introduce a length of time, as in “We need somewhere to rest for two hours.” “We need some time for us to rest” is grammatically possible, but usually unnecessary. “We need some time to rest” already makes it clear that we are the people who will rest. “We need some time for us to rest for” does not sound natural. So the simplest and most natural forms are: “He needs plenty of money to carry out the project.” “We need some time to rest.” The important point is that English does not always require a preposition simply because we can imagine a relationship such as “with money” or “for some time.” The natural structure depends on the noun and the verb being used.