Combination of Conditional Expectations Let $(T,S,\theta)$ be random variables in $\mathbb{R}^3$ with joint pdf noted by $f_{T, S ,\theta}(\cdot)$ 
I want to know if $E[\theta|T\geq t,S\geq s]= \frac{\int_{-\infty}^{\infty} \int_{t}^{\infty}  \int_{s}^{\infty} \theta f_{T, S ,\theta}(t,S, \theta) \;dS \, dT\,d\theta}{\int_{t}^{\infty}\int_{s}^{\infty} f_{T ,S}(T,S)\;dS \, dT}$ can be expressed as a combination of the expectations noted A and B below:
\begin{align} A = E[\theta|T=t,S\geq s]&=\frac{ \int_{-\infty}^{\infty} \int_{s}^{\infty} \theta f_{T, S ,\theta}(t,S, \theta) \;dS \,d\theta}{\int_{s}^{\infty} f_{T ,S}(t,S)\;dS}\\
B = E[\theta|T\geq t,S=s]&=\frac{ \int_{-\infty}^{\infty} \int_{t}^{\infty} \theta f_{T, S ,\theta}(T,s, \theta) \;dS \,d\theta}{\int_{t}^{\infty} f_{T ,S}(T,s)\;dT}
\end{align}
I have in mind (though cannot show) something like a linear combination of A and B, for example, if one assumes that $(\theta,T,S)$ follow a multivariate Gaussian distribution
 A: No it is not. The problem is with notation, and with how things change when two or more rv's are present, compared with the univariate case. When only one rv is present then using the standard conditioning notation "|" to indicate truncation of the domain is ok, because, exactly, no other rv is present. 
Consider now two rv's $X$ and $Y$ with pdf's $f_X(x)$ and $f_Y(y)$, and cdf's $F_X(x)$ and $F_Y(y)$, ranging both in $(-\infty, \infty)$. 
The expression $E(X\mid Y \ge a)$ always means
$$E(X\mid Y \ge a) =\int_{-\infty}^{\infty}xf_{X|Y}(x|y\ge a)dx=\frac {1}{f_Y(y\mid y\ge a)}\int_{-\infty}^{\infty}xf_{XY}(x,y\ge a)dx$$ 
and it is verbally described as a conditional expected value - it is a mistake (confusing) to call it "truncated".
Also, as you can see, $Y$ is not integrated out, and it shouldn't: conditional expected values are a function of the conditioning variable.
Now, as far as I can tell, your RHS is a legitimate expression that is described as "the expected value of $\theta$ w.r.t to the joint distribution with truncated $T$ and $S$"  - but  I don't think that we have a short-hand notation for that... perhaps, something like
$$E_{\theta \tilde T\tilde S}(\theta) ??$$  
where we use the subscript in $E$ to denote that we apply the joint density, and the tilde denote the truncated versions of $T$ and $S$? In any case, don't go with the "conditional" notation, people will understand something else.  
