First of all, note that "synonym" can have two meanings. Paraphrasing a dictionary definition, it can be either a word that's exactly the same or almost the same. This fact is sometimes ignored.
You asked the question:
Should either "moreover" or "also" be removed?
with regard to the following examples:
My town is bad because X and Y, and moreover, the cost of living
therein is also so exorbitant that...
The university itself, moreover, is also unable to launch a serious
defence of the proposed centre.
"Furthermore, besides food intake, other factors can also affect stool
The definitions of "moreover" and "furthermore" tell us that they refer back to a previous thing said:
in addition to what has already been said;
In the third example the sentence begins with "furthermore", so we have no idea of what has already been said. We can assume that it's a statement about food intake being a factor, but we don't exactly have this information.
However, "also" can be seen technically as a redundancy because you can remove it without changing it's meaning:
Furthermore, besides food intake, other factors can
affect stool size.
In the second example the only words before "moreover" are "The university itself", whose meaning is hard to identify exactly, probably because it isn't a complete sentence. Even if we replace "moreover" with "in addition", which is given as a synonym of "moreover", we need to know "in addition" to what? The "what" can't be "The university itself", because it's describing the "university itself" in some way in addition to something else, presumably said beforehand. So I'd say in the second and third examples we just don't have enough information with regard to "Furthermore" and "moreover".
In the case of your first example, we have more relevant information but suffers from a slight ambiguity in exact meaning. We have something basically like:
My town is bad because X and Y, and moreover (in addition to that previously said), the cost of living therein is also so exorbitant that...
To be accurate, we don't know whether the word "moreover" (in addition to) refers to the town being said to be bad, or "X and Y" (the reasons why the town is bad). I believe before we make a judgement as to what is or isn't redundant we should know exactly what the heck the sentence is saying, and it's not entirely clear in all of the example sentences.
Instead I'll focus on a simplified example based on the university example:
The university's partners are unable to fund it. The university itself, moreover, is also unable to fund it.
In this example you can make a case that "moreover" and "also" are mutually redundant. That is, you can remove "also" or "moreover" without having a change of meaning.
Now let's take another version:
The university's partners are unable to fund it. There are many obstacles preventing progress and this is one of them. Moreover, the university itself is also unable to fund it.
Here one interpretation is that the "Moreover" means "in addition to the one obstacle preventing progress" and the "also" is including the university proper among the list of entities that are unable to fund it. They can refer to different things, and I would argue are not redundant with respect to each other necessarily. Of course you could argue "also" is redundant on a very pedantic basis, because:
John is sick. Mary is sick. All the rest are sick.
is equal to
John is sick. Mary is [also] sick. All the rest are sick.
(also is not strictly needed)
Unless you REALLY wanted to get pedantic and say that by the time you read "Mary is sick" John has recuperated and is no longer sick. Haha.
So my opinion is that although in all your examples some redundant words seem to be used, it appears there isn't enough information or context, at least with two of your examples, to make this a certain evaluation.
Should you avoid so-called redundancies or tautologies at all costs?
I would argue there's a place for genuine redundancies, and that it's a stylistic choice. I would definitely use "and also" when just "and" would suffice. Here are examples of redundancies and what many consider to be redundancies:
aid and abet (mean the same thing, exists for traditional reasons).
In idiomatic language
I fell / I fell down (would you fall up? OK this is a phrasal verb, but still, is it necessary?)
Above and beyond
More contentious ones (in my opinion):
* we can not dedicate, we can not consecrate, we can not hallow, this ground.
(consecrate and hallow I'd argue are extremely close to being the same thing.)
* I have nothing to offer but blood, toil, tears and sweat. (Were these not metaphors, then they'd be different things. However being metaphors for sacrifice (or similar), then they all seem to mean the same thing in my opinion.
In literature and poetry
Only this and nothing more.
Syntactic redundancies (or at least strictly unnecessary word addition):
I entered (into) the room
I know (that) it's true
(Some of the above examples are taken from the pleonasm Wikipedia article).