Relationship between Coordinate Time and Proper Time

Everytime I try and think of time dilation, length contraction or any other strange phenomenon predicted by this strangely beautiful theory I get confused!! Luckily we have a metric to do all the thinking for us. In coordinates $x^\mu=(ct,x,y,z)$ with spacetime signature $(+1,-1,-1,-1)$ the metric is given by \begin{equation} c^2d\tau^2 = ds^2 = c^2dt^2 - d\vec{r}^2 \end{equation} where $d\vec{r}^2=dx^2+dy^2+dz^2$. If the coordinates are parametrised by $\tau$ so that $t=t(\tau), x=x(\tau), y=y(\tau)$ and $z=z(\tau)$ then we may write the above equations as \begin{equation} d\tau = \sqrt{dt^2-d\vec{r}^2} \end{equation} which is equivalent to \begin{equation} d\tau = dt\sqrt{1-v^2} \end{equation} where we adopt a timescale for which $c=1$ and $d\vec{r}/d\tau$ is equvialent to the velocity and hence the relation between coordinate time and proper time between two events at $t_1$ and $t_2$ is \begin{equation} \tau = \int_{t_1}^{t_2}\sqrt{1-v^2}dt \end{equation}

To answer your question, a spacetime interval $ds^2=d\tau^2=-g_{tt}dt^2$, can be expressed as \begin{equation} d\tau=\sqrt{-g_{tt}}dt, \end{equation} by definition. Your definition of the spacetime interval $ds^2$ is slightly off, it should read $ds^2=d\tau^2 = -g_{tt}dt^2+...$