Consider a linear time-invariant (LTI) system $$ \dot{x}(t) = Ax(t) + Bw(t); \quad x(0)=x_0 $$ where $x(t)\in\mathbb{R}^n$ is the state, $x_0\in\mathbb{R}^n$ is the initial state, and $w(t)\in\mathbb{R}^m$ is the disturbance. The matrix $A\in\mathbb{R}^{n\times n}$ is Hurwitz, i.e., $\text{eig}(A)\subset\mathbb{C}_{<0}$, and $B\in\mathbb{R}^{n\times m}$. Note that $$ x(t) = \exp(At) x_0 + \int_0^t \exp(A\tau)Bw(t-\tau)d\tau $$ where $\exp(\cdot)$ is the matrix exponential function.
Since $A$ is Hurwitz, there exists $\lambda>0$ such that $\|\exp(At)\|\leq e^{-\lambda t}$. Thus, $$ \begin{array}{ccl} \|x(t)\| &\leq & e^{-\lambda t} \|x_0\| + \|\int_0^t \exp(A\tau)Bw(t-\tau)d\tau\| \\ &\leq & e^{-\lambda t} \|x_0\| + \int_0^t \|\exp(A\tau)Bw(t-\tau)\| d\tau \\ &\leq & e^{-\lambda t} \|x_0\| + \int_0^t \|\exp(A\tau)B\| d\tau . \|w_{[0,t]}\|_\infty. \end{array} $$ This is from Chapter 2 of the book "Feedback Control Theory" by Doyle, Francis, and Tannenbaum.
Question 1: What is the value of $\int_0^t \|\exp(A\tau)B\| d\tau$?
Question 2: Can it be characterized in terms of $\mathcal{H}_2$/$\mathcal{H}_\infty$ gains?
Question 3: Does it have a relation with the controllability gramian $\int_0^\infty \exp(A\tau)BB^T\exp(A^T \tau) d\tau$?
Question 1. This depends on the values for $A$ and $B$ but it is difficult to compute in general. A very conservative upper bound can be computed using the fact that $\|\exp(A\tau)B\|\le \|\exp(A\tau)\|\|B\|$ and that $\|\exp(A\tau)\|\le \exp(\|A\|\tau)$. In this case, we have that
$$\int_0^t \|\exp(A\tau)B\| d\tau\le\dfrac{\exp(\|A\|t)-1}{\|A\|}\|B\|.$$
Note, however, that this bound grows without bound as $t$ increases while the integral converges to a finite value. A better bound can be obtained using the matrix measure (or logarithmic norm)
$$\mu(A)=\lim_{h\to0^+}\dfrac{||I+hA||-1}{h}$$
which verifies
$$\|\exp(A\tau)\|\le\exp(\mu(A)\tau),\ \tau\ge0$$ and yields
$$\int_0^t \|\exp(A\tau)B\| d\tau\le\dfrac{\exp(\mu(A)t)-1}{\mu(A)}\|B\|.$$
Question 2. No. This is more related to the concept of input-to-state stability and the $L_\infty$-induced norm.
Question 3. We have that
$$\int_0^t \|\exp(A\tau)B\| d\tau=\int_0^t \lambda_{max}[B^T\exp(A^T\tau)\exp(A\tau)B]^{1/2} d\tau.$$
However, the integral does not commute with the eigenvalue operator and the square root. So, there is no direct relation with the controllability Gramian.