(1) Here 是 is NOT a copula, but a marker for the cleft sentence, equivalent to the English it-cleft. It emphasizes what comes after.
Historically, the evolution of copula in Chinese started from existting (early form is 惟 and its variations), to nonexisting (者..也 is used instead of a copula) to existing (是 developed into a copula). It's related to cleft sentences since the cleft marker is a further grammaticalization of the copula (happened with 惟 in transition to the second stage). Now 是 is repeating what was happening with 惟. Along the process the cleft or pseudo-cleft sentences get more and more popular. There is an excellent source of reading, 《上古汉语语法纲要》. It is a bit difficult but very illuminating, and it also compares Old Chinese with Middle and Modern Chinese.
Normally 是 can be put at many positions depending on what to emphasize.
(The quoted examples are taken from 今文《尚书》“惟”字分裂句初探, 郭维茹 2016)
It can be used to emphasize the subject,
是老李昨天买了这本书。It was John who bought the book yesterday.
It can be used to emphasize the adverbial,
老李是昨天买了这本书。It was yesterday that John bought the book.
It can be used to emphasize the predicate,
This one is more subtle. Depending on the context, it can emphasize either the verb, the object, or the whole predicate. Compare
“餐厅是在周边一个靠海的小城市。” is a locative sentence, making it only possible to emphasize the subject or the whole predicate with 是. Your example emphasizes the whole predicate, located at one of the surrounding small close-to-see cities. To emphasize the subject, we can say 是餐厅在周边一个靠海的小城市。It's more natural in comparison, e.g. 是餐厅在周边一个靠海的小城市, 不是邮局。(as contrastive focus)
(2)周边 modifies 小城市, not the NP 一个靠海的小城市. The structure is common. For example, 我出生在鄱阳湖边一个美丽的村庄里。In this one and your example, the location of numeral + classifer cannot be switched. It's likely because that the location 周边、鄱阳湖边 is tighter with 在.
Sometimes we can switch the location of numeral + classifer. For example, 昨天来的几个朋友 vs. 几个昨天来的朋友. There are some subtle difference between them. The focus and stress are different: the former on 昨天来的 and the latter on 几个. This is again more meaningful and easier to see if it's acted as a contrastive focus.
(3)No. 周边的一个靠海的小城市 is bad style. For a native speaker the structure of the original sentence is clear, and the wording is neat. Yet, there is a way I think to make it clearer to a learner.
We can add 里 at the end: 餐厅是在周边一个靠海的小城市里。It doesn't change the meaning or focus, but make it more colloquial. It might help your understanding since often what we see is a circumposition rather than a preposition, like 在...上, 从...起. The using of 在...里 clearly segment the sentence into 餐厅 是 在...里.
Or we can make the modifier of 周边 explicit. For example, 餐厅是在我家周边一个靠海的小城市里。Not sure if this helps understanding.
Those being said, 周边的一个靠海的小城市里 is not completely useless. If the speaker pauses to think, then he might add a 的 after 周边 before or while (a prolonged sound of 的) he thinks.