For sentence (1), either singular or plural could be used. It does not matter that "people" earlier is plural, because "people" is actually not the subject of that verb (it is only part of a sub-clause used to set the context for the main sentence). The core of the main sentence is actually:
Wearing a medical mask in the community is not recommended.
Here, the main verb is "is (not)", and the subject of the sentence is actually "wearing a medical mask (in the community)". This is a gerund phrase referring to the activity of a hypothetical "someone" wearing "a medical mask". Since the hypothetical "someone" can be a single person (i.e. any individual from the group of "people without respiratory symptoms"), it can all be expressed in the singular, and it works fine.
Alternately, one could also talk about the activity of some hypothetical "group of people" wearing "medical masks", which in this case doesn't change the meaning very much, and still makes sense, so you could also phrase the whole thing in the plural and it works either way.
But the important point here is that the "people without respiratory symptoms" is not actually technically the subject of "wearing" here.
Sentence (2), on the other hand, is just plain wrong. It is grammatically incorrect (you're right that it should be plural, because "spine has been ..." is a relative clause attached to the plural noun "patients", and thus needs to agree with that noun).
Sentence (3) is, I believe, also incorrect. Rather than singular/plural, the sentence you quoted appears to actually be trying to use these terms as uncountable nouns. It is common to refer to diseases (such as COVID-19) as uncountable nouns ("they all had COVID-19"), but not symptoms such as "fever" or "cough", etc. Those should be either plural "fevers"/"coughs"/"sore throats", or singular "a fever"/"a cough"/"a sore throat".
Note, however, that when talking about people posessing things ("have a fever", "have a dog", etc), if you have a plural subject you can actually use either a plural or singular object ("they have fevers"/"they have a fever"). The meaning is sometimes slightly different between the two because using the singular form implies that each individual in the group has one thing: "they all have dogs" suggests that each person has at least one dog, but may have more, or possibly that there are a bunch of dogs shared between them, but "they all have a dog" suggests that each person has exactly one dog. In the case of "a fever", etc, each person can only have one at a time anyway, so the two meanings are the same.