Learn English – “longer”, “for longer”, or “more”

While those might mean the same for the laymen, from a medical point of view, there is a difference between illness and sickness.

Medical sociology has long made the distinction between illness and sickness. Illness is the objective diagnosis that an external impartial observer is able to make based on the constellation of symptoms which the patient presents. Sickness is the social role that the patient adopts as the patient and other concerned stakeholders, in relationship with the patient, interpret the meaning of the illness.

From what I get of it, someone might see themselves as sick (with the social/role aspect of it) but not actually be ill (in a medical sense). Also, this paper might provide some useful reading.

To some extent, this is a general reference question because there are numerous pages on the Internet that discuss adverb place, e.g., this one and this one. However, the question is still an interesting one because there are frequent disputes about specific placements of specific adverbs and whether different placements change the meaning of specific sentences.

The first problem with what you're taught about where adverbs should be placed is that it depends upon the type of adverb and, sometimes, on the specific sentence and specific context of the sentence. For example, most native speakers of English would argue that the following two sentences mean exactly the same thing:

I only eat ducks.
I eat only ducks.

In these two sentences, where the adverb is placed is more or less a matter of style (personal preference). But if the sentences were modified to read, for example:

I only eat ducks, not shoot them.
I eat only ducks, not shoot them.

the syntax of first one makes more sense to me. I think the style is better. But that may be idiosyncratic.

However, assuming two witches chatting online in a contemporary remake of the Hansel and Gretel:

{1} I only eat children, because adults taste bitter. [NB: comma is optional]
{2} I eat only children, because adults taste bitter. [NB: comma is optional]
{3} I eat children only, because adults taste bitter. [NB: comma is not optional]

Only the third one makes instant sense to me. Most native speakers would probably say that the first and third are semantically equal and that the second is ambiguous because it isn't clear whether only is being used as an adverb or an adjective here.

The third sentence could also be:

{4} I eat children, only because adults taste bitter. [NB: comma is not optional]
{5} I eat children only because adults taste bitter.

in which case the meaning of {4} and {5} are different from {1}-{3} and different from each other. Sentence {4} emphasizes why the witch eats children, and sentence {5} is ambiguous: Does only function as an adverb or and adjective here?

The second problem is that when an adverb is placed at the end of the sentence, it usually modifies the entire sentence, just as when it is placed at the beginning of the sentence. In such cases, however, it's usually set off by a comma. But this is far too extensive a topic to discuss in a single answer here.

Of the example sentences you list, I think I can quickly run isn't really acceptable even though it's grammatical and meaningful. It just doesn't sound natural to me, except, perhaps, for a song.

The third problem is that your example contains only adverbs of manner (-ly adverbs). Other adverbs have different rules. It would be better to provide specific sentences, especially longer and more complex sentences. Then the question will become more meaningful for more users.