The short answer to your title question is, 'No.'
A single metaphor cannot be a mixed metaphor.
If you take the form of a metaphor to be (a = not a), mixing two unrelated concepts, eg 'He's a snake' referencing a human being, is part of the process.
If you say, in the context of introducing a new joiner to the personalities in an organisation, 'Watch out for that snake in the grass', where a person is a snake and the workplace is either a jungle, or an arcadian vale, depending on your chosen context, then you are grouping related metaphors.
If you say, 'We were sailing on fine until we hit some roadworks', then you're mixing metaphors - the process was a journey by ship, and suddenly (this is where the mixing referred to in the phrase 'mixed metaphors' happens) your process becomes a road trip - when exactly did the ocean change into Route 64?
So the M-W definition is correct.
As for your second question, which references a 'second sentence' (understood as 'man is lion and tiger', as 'lion is tiger' is a single metaphor, therefore would not fit the mixed model), you're raising a new distinction.
If I continued my introductory tour and said, 'At work, he's a snake, but in a battle, he's a lion', the shift is deliberate, and would not be classed as a mixed metaphor.
So, the differentiator here seems to be linked to whether the juxtaposition of ideas is conscious or unconscious, and if conscious, for what effect.
In most cases, a mixed metaphor will arise through an unconscious process. That, however, can be replicated consciously, as we are doing here, for educational and informational purposes.
I think it's safe to say that in its natural habitat, a mixed metaphor is a kitten playing with a ball of wool unaware it's in a saucepan of boiling water. In a laboratory, it is definitely aware, and wants out!
ADDITION:
In 'The Language of Metaphors', Andrew Goatly talks about a vehicle or V-term as being part of a metaphoric construction.
He uses the following sentence as an example:
“The past is a foreign country; they do things differently there.”
the Vehicle-term or V-term is “a foreign country”, the Topic-term or T-term is “the past”, the Ground or G-term is “they do things differently”. (p 8)
He looks at the link between syntax and metaphor, pointing out that:
Lakoff and Johnson (1980) fail to take [syntax] into account when
discussing metaphorical mixing. They claim that what makes the
following examples of mixing permissible and impermissible
respectively is that the two metaphorical schemata involved do or do
not share entailments.
(1) At this point our argument doesn’t have much content. (Lakoff and Johnson 1980:92)
(2) The content of the argument proceeds as follows, (ibid.: 95)
It seems equally likely that it is the intimacy of the syntactic bond between subject
and verb in (2) which makes for a sense of mixing, whereas (1) only has a distant bond
between prepositional complement and object of the same clause (see also Low 1988:132).
Both examples involve 2 distinct metaphors: (1) an argument is a container; an argument is situated on a 'continumm'. (2) an argument is a container; the contents are capable of movement (the dissonance, for me, is that it is the argument, as container, which is more likely to move, rather than the contents).
In his analysis of diversification, Goaty notes:
Diversification rules out any syntactic relation between the two different V-terms, which > would amount to Mixing.
(p 262). Again, note the 'two different V-terms' requirement. He goes on to define mixing as follows:
Mixing occurs when two V-terms are put into a syntactic relationship
with each other, while their conventional referents, the Vehicles, can
contract no such corresponding relationship to each other in the
world...
(p 263). Under 'Mixing', he notes:
There are two factors, therefore, which are relevant to the perception
of metaphorical Mixing. One is the strength of syntactic bonding and
syntactic proximity between the two V-terms. The other is the degree
of activity of the metaphor.
(p 277 f). He analyses the following sentence:
the sky was teeming and tearing along, a vast disorder of flying shapes and darkness and
ragged fumes of light
as follows:
In the phrase “ragged fumes of light” two incompatible V-terms have
close syntactic bonds, i.e. the relation of noun head and premodifier.
“Ragged” should be applied to a flexible solid substance of some kind,
typically cloth. “Fumes” must apply to a gaseous substance.
Again, you have more than one metaphor combining to create a mixed metaphor:
Sky is cloth - Sky is gas.
Goaty goes on to point out that in the larger passage, which I've only quoted the relevant part of here:
... the Mixing and the bombarding with disparate Vehicles is extreme.
However, we sense that the disorder being described by Lawrence here
is the very motivation for this metaphorical Mixing—to mirror a rapid
succession of physical impressions and their accompanying emotions.
All this simply confirms what I've said - but as you asked for a second opinion, it is hereby provided.
Best Answer
The literal meaning of running a mile a minute seems irrelevant to the action portrayed in the panel; instead the metaphorical meaning suggested in the question (that Captain America has been working to the point of exhaustion) is quite plausible and is reinforced by several pictures in the panel. In addition there is no obvious and plausible meaning for “running a mile a minute” that is risqué, bawdy, or ironic.
In short, it appears that “running a mile a minute” has been used metaphorically, and that whoever suggested it was a double entendre apparently doesn't know what one is: “A phrase that has two meanings, especially where one is innocent and literal, the other risqué, bawdy, or ironic; an innuendo”.