Let's take these one at a time, but first, a little review of future tense in English.
Unlike past tense, and present tense, which have four forms each in English, as follows:
- past/present simple; e.g. "ate/eat"
- past/present perfect; e.g. "had/has eaten"
- past/present continuous; e.g. "was/is eating"
- past/present perfect continuous; e.g. "had/has been eating"
...future tense has at least seven forms, including all four of these (present tense plural forms) with the addition of "will," e.g. "will have been eating," and these additional forms:
- simple present tense; e.g. "He works tomorrow."
- present continuous tense; e.g. "He is working tomorrow."
- "going to"; e.g. "He is going to work tomorrow."
Notice that every one of these three additional forms uses a present tense verb without "will." Whenever one of these three forms is used, context determines that it is future tense: in the example sentences here, the word "tomorrow" clearly indicates a future time.
Typically English speakers avoid the repetition of "will" in a future context, and use present tense verbs in its place for additional verbs in the sentence.
For example: "I will have traveled for four hours by the time I arrive at the airport tomorrow."
In this example, saying "will arrive" is actually incorrect, grammatically or idiomatically. This is often the case with future verbs indicating a sequence of future events: "will have traveled" being the future perfect, and "arrive" being future simple--represented in present simple form.
Now, let's consider your examples.
Tomorrow at 10 o'clock, there will be something that comes up.
Tomorrow at 10 o'clock, there will be something that will come up.
Notice that neither of these sentences provides a sequence. The expression allows for each verb to be acting independently of the other, via a separate, dependent, clause. This means that a repetition of "will" is possible. However, as a repetition of "will" is typically avoided by English speakers, Example 1 is more idiomatic (natural). The use of "will" for the second verb, i.e. "will come up," is considered redundant and unnecessary, and, in this case, makes no difference to the meaning of the sentence. The second inclusion of "will" does not change the time to which its verb is applied.
Now, to your second example.
Tomorrow I will fix the errors that the computer shows.
Tomorrow I will fix the errors that the computer will show.
This example differs in that the context of the second verb is ambiguous. It could be understood, in Example 3, as applying either to the present time, or to the future--because a present simple tense form of the verb can represent either present or future. If the computer is showing those errors right now, and you intend to fix them tomorrow, or if the computer will show the errors tomorrow, and is not showing them yet, is unclear. Example 4 removes this ambiguity, and plainly indicates that the errors are expected tomorrow; therefore, in this case the repetition of "will" is acceptable.
I will eat food that they provide.
I will eat food that they will provide.
These two sentences are very nearly identical, but not exactly. The use of present simple form for "provide" in the first sentence gives it a factual tone, as if it is certain. While "will provide" in the second example can also be factual, the addition of "will" implies that the food has not yet been given, as this is still clearly future, and therefore a greater measure of faith that it will be given is implied. Example 5 assumes that the food is being supplied as a matter of course, and will be supplied without question; whereas Example 6 indicates that the food has not yet been provided, but is merely expected or planned. Again, due to the potential difference in nuance of meaning here, the additional "will" is grammatically permissible.
Wherever it would make no change to the meaning, additional instances of "will" beyond the first one are considered redundant, and are usually omitted. In a verb sequence, where two or more future times are indicated (e.g. future perfect + future simple), only one "will" is grammatically permissible. However, when the addition of "will" clarifies an ambiguity with respect to present tense usage versus future tense usage, it is permissible to include it, and it does, in such a case, affect the meaning of the sentence.