Do I condition a constant while computing the conditional expectation? I have this event $A$ and I know $P(A)$.
I also have a r.v. $T$, which is exponential with a given $\lambda$. 
I want to compute ${\bf E}[T+5|A]$.
I remember that unconditionally, ${\bf E}[T+b]={\bf E}[T]+b$. But does it mean that ${\bf E}[T+b|A]={\bf E}[T|A]+b$ or do I have somehow to condition the constant $b$ as well?
Conditioning a constant seems weird to me, but on the other hand I have a hunch I'm missing something. 

Yes, the Linearity of Expectation holds for a conditional expectation too.
When $T$ is a random variable, $A$ is an event, and $b$ a constant, then:  
$$\mathsf E(T+b\mid A) ~=~ \mathsf E(T\mid A)+b$$
