I have a bit of a struggle with the に particle. Could somebody explain to me why it is used here?
結婚のお祝いに、部長に時計をいただきました。 (first に)
In this case, ni is indicating the time at which the speaker received the watch. This is similar to the preposition "at". You can roughly translate the sentence as "I received this watch from the manager at my wedding celebration".
I don't know any formal, textbook-like definition to this, but my intuition tells me ~~ ni benri means more or less "convenient for ~" because there are several phrases in Japanese that use adjectives with ni like this. Consider the following example:
p 乗の対数は、対数の p 倍に等しい。 (Source: Japanese Wikipedia article on logarithms)
A logarithm to the p-th power is equivalent to the logarithm times p.
You can also translate the phrase as "equals the logarithm times p", but I hope you can feel something about these patterns.
In this case, ni is marking purpose and functions similar to the phrase "in order to ~." The phrase in your example means "(in order) to repair the car". You can translate the sentence as "It will take one week to repair the car".
There are two common patterns in which ni is used like this. The first one is with the so-called "masu-stem" of a verb when indicating the purpose of a motion verb. The other one is usually with a nominalized verb phrase and looks like noni (don't confuse it with the other noni!) Here are some examples:
I'm planning on going out to eat with my girlfriend
このレポートを書くのに一ヶ月かかりました。 (Source: A Dictionary of Basic Japanese Grammar, entry titled "のに(2)" )
It took me a month to write this paper.
~ ni suru is an extremely common pattern in Japanese and usually translates as "decide on ~". It has many other usages (e.g. koto ni suru, ~ wo ~ ni (shite), etc.) and there should be many questions regarding ni suru on this website, so I will not explain it here. This example can be translated as "Let's have sukiyaki for tonight's side dish."
You can think of this usage of the ni particle as an indirect object marker. This is roughly equivalent to several prepositions in English such as "to" and "in". Again, there should be many questions on this website that describe this usage of ni, so I will not explain it here.
Notice the verb is 逢う au, which means "to meet" with a nuance of something bad and undesirable. The highlighted phrase in this example can be translated as "not get into an accident"