In English, adverbs in -ly are often, not always, very mobile in a sentence.
In the sentence, "I ran to the station." if I add "quickly" there are 4 possible places where I can insert "quickly".
Quickly, I ran to the station.
I quickly ran to the station.
I ran quickly to the station.
I ran to the station quickly.
How mobile is 快速 (or other word for quickly) in this sentence? Can 快速 appear anywhere else in this sentence?
我快速跑向车站。
A general rule of thumb is to modify directly before what you want. This is least likely to make mistakes.
Also, 的地得 words bind "tighter", and is nearly always used directly before what it modifies.
Examples: (Parens mark what is modified, square brackets mark the modifier)
I am hard at learning Chinese.
我正在【努力地】（学习）中文。
No： 我正在学习努力地中文。（You never modify the noun with the adverb）
No： 我努力地正在学习中文。(Bind tighter)
Sometimes okay: 努力地我正在学习中文。(Not modern Chinese, but may be used literarily)
The last one is not similar to "Under the tree stands a girl", you stress Under the tree in the English version. In the Chinese version you can stress when you read it out, or just make 努力 into a separate sentence: 我在学习中文，并且我非常努力！
I think 快速地我跑向车站 is acceptable, but sounds a bit weird and even hilarious, since it reminds me of a famous poem《再别康桥》. It might be common in literature. 我跑向快速地车站 is absolutely wrong. 我跑向车站，快速地 is sort of similar to the first case: acceptable but too literary. 我跑向车站，健步如飞 is more natural, although I don't know if 健步如飞 functions as an adverb here.
The verb 是 seems special. Any of the following makes perfect sense.
This is obviously wrong.
这是显然错的
这显然是错的
显然这是错的
这是错的，很显然
When you say, 我快速跑向车站, the complete sentence is 我快速地跑向车站。
Whenever there is a 什么什么地 used as an adverb, the only correct place is in front of the verb.
You have options to use other kinds of sentences to express similar meanings. But the difference could be subtle.
In English, the sentence "Quickly, I ran to the station" and "I quickly ran to the station" are very close, but if you translate them "directly" to Chinese with the same word order, the meanings can be altered.
"快速地，我跑向车站" is OK only in a poem, such as 轻轻地，我将离开你. It is not popular to use this kind of "inversed word order" in daily life.
"很快地，我跑向车站" means "Immediately, I ran to the station (after I got the message)". It is about the whole event (what happened), not only the action (ran).
"我跑去车站，很快" could mean "我跑着去车站，很快就到"。I would run to the station and very quickly I would be there (to see you).
In your particular example, only 我快速跑向车站 is used and can be understood without some spiritual raising of eyebrows.
Before everything: I want to comment on 快速地，我跑向车站 vs Quickly, I ran to the station separately.
Back from digression, in order to generalize what we see in 我快速跑向车站, first, I would observe that modifiers of verb, noun, adjective, are always immediately prepositioned in modern Chinese. Perhaps that's also true for all verb phrases and noun phrases -- however one may discover the plausible phrase structures in Chinese.
See this primer on 副词 for some examples of modifiers (副词 is modifiers for verbs and adjectives). You can discard the claim of the positioning at first and just borrow their examples of 副词; then consider other 副词 they may remind you of. Can you think of examples of post-positioning? I bet it is difficult!
Next, I would observe that 快速 is not a valid modifier for anything in your example sentence 我跑向车站 except either 跑 or 跑向车站.
I am less willing to conclude that there is strict syntactical structure aside from the immediate prepositioning rule for modifiers. But common Chinese sentences are heavily guided what would commonly be semantically consistent. If it is not semantically valid, it will raise spiritual eyebrows reflexively.
That is how I would theorize why only 我快速跑向车站 is grammatical.
As for "快速地，我跑向车站" vs "Quickly, I ran to the station".
When you read "Quickly, I ran to the station", do you really see "quickly" as being associated to just the verb "ran" -- especially when you first read/hear it? Or do you absorb the idea of "quickly" and then allow it to be added to whatever follows would convey?
I would say in any language, you may attempt to extract a modifier phrase (ie. adjective phrase, adverb phrase, preposition phrase) to the front or rear of a sentence and there is a chance that, even though not common used, it is tolerated and even used somewhere, perhaps in poetry. Unless a sentence structure modifier phrase + comma + "verb phrase with noun" is commonplace in a language, I would hesitate to consider it a part of the grammatical system.
The core idea in my answer is that (1) I suggested that there an "immediate prepositioning rule" for modifiers in Chinese. 我快速跑向车站 is an example of that.
But there are some more nuances which may be of interest to you. Listed with (2)-(4) below to facilitate quick reading as this answer is getting very long!
(2) Chinese modifiers often are not morphologically different when applied to noun, adjective, and adverb. As a result, there is a sense that you can flexibly try placing the modifier 快速 in most places in the sentence. However, none can semantically work.
(3) I would suggest that there is an emphasis in the customary association of words that would be semantically sensible to combine. Hence, in the example of 我快速跑向车站, if you try any other order, it would perhaps raise extra spiritual eyebrows, giving an illusion of inflexibility.
Finally, let me touch on the immediacy part. With what we said above, do we really have an example of non-immediacy for adverb in the English examples?
When taken the view of adverbs may modify verb phrases in all of:
"quickly" is either immediate before or after a verb or a verb phrase. So if you consider the adverb as modifying either verb or verb phrase, (4) you never lost immediacy.
I would be intrigued to find examples of non immediate modifiers being commonplace in any language and if someone can comment the examples they know, I would love to be intrigued!
Structurally, they're pretty much just as mobile.
Mutatis mutandis, adverbs with "ly" in English are comparable to adverbs with "de" in Chinese (variously spelled 的, 得 or 地 according to the adverb's position in the sentence). So, to answer your question, the following are indeed all somehow plausible:
Semantically, however, connotations may vary because changing the default position of these adverbs may change what they're intended to modify.
These adverbs are usually intended to modify the verb, so their default position is pre-verbal. Non-default positions defy this syntactic rule, leaving the referent (what the adverb modifies) ambiguous. In most cases, it will still be understood to be the verb. In other cases, the semantics of the adverb allows it to plausibly modify much more than the verb. For example, the adverb in "迅速地把车刹住" refers to the manner in which the subject attends to the whole sentence, whereas in "把车迅速地刹住" the adverb strictly refers to how the main verb is performed. This is sometimes the case in English too. For example, "Hopefully, she will leave" is different to "She will leave hopefully" because the adverb in the former refers to the whole sentence (so we are full of hope that she leaves), whereas in the latter the adverb is only tied to the main verb (so it's her who leaves full of hope).
So, in both English and Chinese, these adverbs are mobile enough that they can occur in non-default positions. But when they do, they will have a wider or more ambiguous scope of reference and, because of this, will often carry a literary/poetic feel or some kind of contrastive focus (especially if placed last).
