A "Completion" of $ZFC^-$
Without a definition of “large cardinal axiom”, I’m going to ignore Q1 and Q3.
The answers to Q2 and Q4 are negative by the following general principle. (For Q4, we take $T_0$ to be the set of $\mathcal L_{\mathrm{set}}$-consequences of $T_1$.)
$\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\Wit{Wit}\let\eq\leftrightarrow\def\gonu#1{\ulcorner#1\urcorner}\let\ob\overline$Proposition: Let $T_0$ be an r.e. theory interpreting Robinson’s arithmetic, and $\Gamma$ a set of sentences for which $T_0$ has a truth predicate $\Tr_\Gamma(x)$, that is, $$\tag{$*$}T_0\vdash\phi\eq\Tr_\Gamma(\ob{\gonu\phi})$$ for all $\phi\in\Gamma$. Then no extension of $T_0$ by a set of $\Gamma$-sentences is a consistent complete theory.
Proof: Let $S\subseteq\Gamma$, and assume for contradiction that $T=T_0+S$ is consistent and complete. The basic idea of the proof is that we can define in $T$ a truth predicate $\Tr(x)$ for all sentences as “$x$ is $T_0$-provable from a set of true $\Gamma$-sentences”, contradicting Tarski’s theorem on the undefinability of truth.
In more detail, we fix an interpretation of, say, $S^1_2$ in $T_0$ so that we have basic coding of sequences of integers, and a (polynomial-time) proof predicate for $T_0$. We define $\Wit(w,x)$ to be the formula
“the sequence $w$ is a $T_0$-proof of a sentence $x$ from extra axioms, each of which is a sentence $a\in\Gamma$ such that $\Tr_\Gamma(a)$,”
and we put
$$\Tr(x)\eq\exists w\,(\Wit(w,x)\land\forall w'<w\,\neg\Wit(w',\gonu{\neg x})).$$
Claim: Whenever $T$ proves a sentence $\phi$, it also proves $\Tr(\ob{\gonu\phi})$. Whenever $T$ proves $\neg\phi$, it also proves $\neg\Tr(\ob{\gonu\phi})$.
Using the Claim, we can easily finish the proof of the Proposition: by Gödel’s diagonal lemma, there is a sentence $\alpha$ such that
$$T_0\vdash\alpha\eq\neg\Tr(\ob{\gonu\alpha}).$$
Since $T$ is complete, it proves $\alpha$ or $\neg\alpha$. If $T\vdash\alpha$, then $T$ proves $\neg\Tr(\ob{\gonu\alpha})$ by the definition of $\alpha$, and $\Tr(\ob{\gonu\alpha})$ by the Claim, hence $T$ is inconsistent, contrary to our assumptions. The case $T\vdash\neg\alpha$ is similar.
Now, to prove the Claim, assume $T\vdash\phi$. We can fix a $T_0$-proof of $\phi$ from some $\psi_1,\dots,\psi_k\in S$, which has a standard Gödel number $n$.
By $\Sigma^0_1$-completeness, $T$ proves that $\ob n$ is a $T_0$-proof of $\ob{\gonu\phi}$ from $\ob{\gonu{\psi_1}},\dots,\ob{\gonu{\psi_k}}$. Moreover, $T$ proves each $\psi_i$, hence also $\Tr_\Gamma(\ob{\gonu{\psi_i}})$ by $(*)$. Thus,
$$T\vdash\Wit(\ob n,\ob{\gonu{\phi}}).$$
On the other hand, let $m<n$, we will show
$$T\vdash\neg\Wit(\ob m,\ob{\gonu{\neg\phi}}).$$
This again follows by $\Sigma^0_1$-completeness unless $m$ is an actual Gödel number of an actual $T_0$-proof of $\neg\phi$ from some sentences $\chi_1,\dots,\chi_l\in\Gamma$. Since $T$ also proves $\phi$, this means
$$T\vdash\neg\chi_1\lor\dots\lor\neg\chi_l,$$
hence
$$T\vdash\neg\Tr_\Gamma(\ob{\gonu{\chi_1}})\lor\dots\lor\neg\Tr_\Gamma(\ob{\gonu{\chi_l}})$$
by $(*)$, hence $T\vdash\neg\Wit(\ob m,\ob{\gonu{\neg\phi}})$ as needed.
Since $T$ knows that the only numbers below $\ob n$ are $\ob0,\dots,\ob{n-1}$, we have established $T\vdash\Tr(\ob{\gonu\phi})$.
The second part of the Claim is similar: assuming $T\vdash\neg\phi$, we fix its standard proof with Gödel number $n$, and we show
$$T\vdash\Wit(\ob n,\ob{\gonu{\neg\phi}})$$
and
$$T\vdash\neg\Wit(\ob m,\ob{\gonu\phi})$$
for all $m\le n$, which implies $T\vdash\neg\Tr(\ob{\gonu\phi})$.