Does this variant of Multiplicative Linear Logic with mix rule enjoy cut elimination?
For question 1, if by "deadlock" you mean "non-trivial vicious circle", then the answer is obviously yes, simply because a non-trivial vicious circle cannot be typed: you will have a cut between a formula $A$ and a formula $F$ containing $A^\bot$ as strict subformula, so it is impossible that $F=A^\bot$. But this is kind of trivial: by refusing a priori to consider cyclic wires as deadlocks, you are eliminating the only possible typed deadlock, so of course you are deadlock-free...

For question 2, there are definitely problems with your sequent calculus.  First, I assume that you are working with orthogonality defined by $(A\cdot B)^\bot=A^\bot\cdot B^\bot$.  Then, you have that the formula $A^\bot\cdot A$ is essentially self-dual and, since it is provable, you have a proof of the empty sequent. But your inference rules (except cut) still verify the subformula property, which means that there is no cut-free proof of the empty sequent. In other words, you are forgetting the rule
$$\frac{}{\vdash}$$
Without this rule, cut-elimination does not hold. There is a further problem though, namely that you cannot represent the cut-elimination steps inside the sequent calculus: in other words, as a type system, you do not have subject reduction.  For that, you need to consider a unary cut rule:
$$\frac{\vdash\Gamma,A^\bot,A}{\vdash\Gamma}$$
Note that, like the tensor rule, the binary cut rule is derivable from this one (and mix).

For question 3, I assume that you are asking whether the system captures your notion of deadlock-freedom.  Since you are excluding cyclic wires a priori (i.e., you are artifically adding a cut-elimination rule that makes them disappear) the the answer is, again, trivially yes: a net normalizes (with your additional reduction rule) to what Lafont calls a reduced net (no vicious circle, no active pair) iff it is typable in the above system. This is provable by standard arguments: subject reduction/subject expansion and typability of reduced nets.

For question 4, I doubt anyone has published anything because this system is so degenerate that I honestly do not see what its interest may be.  In fact, one may further simplify the types by eliminating duality altogether: there is no negation, formulas are just
$$A,B::=\alpha\mathrel |A\cdot B$$
and axioms/cuts introduce/eliminate pairs of equal formulas. This system, albeit logically trivial (because of self-duality), has the same normalization property as above. It boils down to the so-called relational semantics of MLL, which is applicable to untyped structures too, in which tensor equals par.

Regarding your search for a Danos-Regnier correctness criterion for single-agent nets, there is not much to be said: if you make cyclic wires disappear "by fiat", then the self-dual "typing" above suffices and correctness criteria play no role; if you stick to the standard definitions, then there can be no meaningful correctness criterion: every one-conclusion net cut with itself generates cyclic wires.