The answer is yes, there are "general features" of
the immune system and basically of our genome that combat many different pathogens, but also there are very specialized ones, only useful against a few. To make the matter more complicated, recent research has found that the immune system "cross-learns" from infections (and possibly from vaccines as well).
First on "purely genetic" defences, some alleles confer enhanced resistance to some viruses. Quoting a 2007 review
Some proteins that are required for
surviving viral infection are used to combat many other
microorganisms as well. In mammals, for example, myeloid
differentiation primary-response gene 88 (MyD88), a
Toll-like receptor (TLR) adaptor molecule, is required for
effective resistance to herpesviruses, Toxoplasma gondii
and other organisms that have few obvious features in
common. By contrast, some defences that evolved recently
are matched against specific microorganisms and may
operate only within a single host species. For example, the
Ly49H receptor expressed by mouse natural killer (NK)
cells seems to recognize only mouse cytomegalovirus
(MCMV), and does so only in some strains of mice, as the
receptor-encoding gene has been deleted in other strains.
More tentative defences are also apparent. For example,
mutational abrogation of the human CC-chemokine
receptor 5 (CCR5) protein offers strong protection against
infection with HIV, but the most common protective allele
has not been driven to fixation in humans. And it is likely
that many other potential resistance mechanisms remain
to be exploited in mammals.
Some protective antiviral systems are cell autonomous,
whereas others depend on multiple specialized cell types
that interact both with the infected cells within the host
and also with one another. In mammals, the response to
viral infection overlaps substantially with the response
to bacteria, using some of the same sensing, signalling and
effector mechanisms. In insects, there is a greater reliance
on cell-autonomous defence, and new advances must be
made in defining the pathways that are involved. [...]
Resistance to viral infection comes at a definite cost.
In mammals, the cellular systems that confer protection
against viruses are also capable of causing autoimmunity. This may be seen as a reflection of three facts. First, viruses
have imposed a need to distinguish between host nucleic
acid and foreign nucleic acid — a challenge that has been
met by mammals in the large part, but not completely.
Second, viruses certainly contributed to the evolution of
adaptive immunity, upon which autoimmunity is predicated.
And third, some of the elements of innate immunity
that support autoimmune disease (among them the
|IFNs and cells that produce them) evolved largely to
Beyond this "fixed"/innate defense system, there's another more recently discovered one that can be considered "intermediate" (i.e. between innate and the classical def of "adaptive immunity"):
Protection against reinfection has been reported not only in plants and invertebrates that do not have adaptive immunity (4), but also in mammals, with old and new studies demonstrating cross-protection between infections with different pathogens (5). These studies have led to the hypothesis that innate immunity can be influenced by previous encounters with pathogens or their products, and this property has been termed trained immunity or innate immune memory.
(In fact this discovery has led some to propose a reconsideration of the immune system dichotomy.)
Also, you assume (in your #2) that COVID-19 outcomes are always better with more immune response, but that might not actually be the case (harking back to a hypothesis about the 1918 flu deadliness):
Some of the earliest analyses of coronavirus patients in China suggested that it might not be only the virus that ravages the lungs and kills; rather, an overactive immune response might also make people severely ill or cause death. Some people who were critically ill with COVID-19 had high blood levels of proteins called cytokines, some of which can ramp up immune responses. [...]
A combination of damage from both a virus and the immune response to it is not uncommon, says Rafi Ahmed, a viral immunologist at Emory University in Atlanta, Georgia. The effects of 'hit-and-run' viruses such as norovirus, which make people sick almost immediately after infection, are more probably due to the virus itself, he says. By contrast, people infected with viruses such as coronavirus do not show symptoms until several days after infection. By then, collateral damage from the immune response often contributes to the illness.
Based on the convergence of epidemiological data and animal research, the auto-immune response is suspected to have been a (probably more significant) factor in 1918:
unlike contemporary influenza strains, which typically affect the very young and the elderly most severely, the 1918 influenza pandemic was mostly fatal in young adults, who generally possess more robust immune systems.
The work of Kobasa et al. substantiates the findings of Kash et al., who showed in mice that the 1918 virus triggered a vigorous innate immune response that was linked to fatalities. Although the mechanisms of tissue destruction were not addressed in either study, the work clearly demonstrates the vital function of early innate immune defences in controlling the virus. It seems that the pandemic 1918 virus had a genetic composition and rapid replication kinetics that may have resulted in an excessively vigorous innate immune and inflammatory response that contributed to severe tissue damage, disease and death.