Properties of the time integral of Wiener process

As an integral of a zero-mean Gaussian process, your $X_T$ is a zero-mean Gaussian process as well. Its covariance function can be calculated via

$$ c(s, t) = \int^s_0 \int^t_0 \min(u, v) \; du \; dv \; , $$

which yields

$$ c(s, t) = \frac{\min(s, t)^2}{6} \left( 3 \max(s, t) - \min(s, t) \right) . $$

In terms of sample functions, your expectation is correct. Being an integral of a random process with (almost surely) continuous sample paths, it indeed has (almost surely) differentiable sample paths.

It does not have independent increments, as discussed here.