Constants of motion from a Lagrangian

If $q_i$ denotes the generalised coordinates, then note that a time translation $t \to t+\epsilon$ infinitesimally corresponds to $q_i \to q_i + \epsilon \frac{d}{dt}q_i$ and so $\delta q_i = \dot{q}_i$. The change in the Lagrangian is,

$$\delta L = \frac{\partial L}{\partial q_i} \dot q_i + \frac{\partial L}{\partial \dot q_i} \ddot q_i = \frac{d}{dt}L$$

which is the total derivative if $L$ has no explicit time dependence. By Noether's theorem, we have a conserved quantity,

$$Q = \frac{\partial L}{\partial \dot q_i}\dot q_i - L $$

which we can recognise as the Legendre transform of the Lagrangian, that is, the Hamiltonian and thus the energy of the system is conserved. Clearly this applies to your Lagrangian.

As you noted, your Lagrangian is also invariant under $\theta \to \theta + \alpha$ for which $\alpha \delta \theta = \alpha$, and $\delta L =0$. We thus have a conserved quantity which we can identify as the conjugate momentum, namely,

$$p_\theta = \frac{\partial L}{\partial \dot \theta} = \frac12 mr^2.$$

Another way to see this is that the Euler-Lagrange equations read, $\frac{\partial L}{\partial \theta} = \frac{d}{dt}\frac{\partial L}{\partial \dot \theta}$ and since there is no $\theta$ dependence,

$$\frac{d}{dt}\frac{\partial L}{\partial \dot \theta} = \frac{d}{dt} p_\theta = 0.$$

In your case, the answer is yes, it's that simple. The reason is that all the constants of motion you're dealing with are related to some translational symmetry. Energy corresponds to translations in time, angular momentum to rotations (translations in $\theta$) and linear momentum to translations.

There's a theorem that relates symmetries of the lagrangian and constants motion: Noether's theorem. It says that for any symmetry there's an associated constant of motion, which can be calculated using the lagrangian and the symmetry transformation. That's a general way of computing constants of motion.

For time-independent lagrangians, a symmetry (a transformation $x\mapsto x(\lambda)$ such that $L(x(\lambda),\dot{x}(\lambda))=L(x,\dot{x})$, where $x$ denotes collectively all coordinates) implies a constant of motion \begin{align} C=&\sum_i\frac{\partial L}{\partial \dot{x_i}}\frac{\partial x_i}{\partial\lambda}. \end{align}

A simple derivation (without too much detail) is \begin{align} \frac{dC}{dt}=&\sum_i\left[\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x_i}}\right)\frac{\partial x_i}{\partial\lambda}+ \frac{\partial L}{\partial \dot{x_i}} \frac{d}{dt}\frac{\partial x_i}{\partial\lambda}\right] \\ =& \sum_i\left[ \frac{\partial L}{\partial x_i}\frac{\partial x_i}{\partial\lambda}+ \frac{\partial L}{\partial \dot{x_i}} \frac{\partial \dot{x}_i}{\partial\lambda}\right] = \frac{dL}{d\lambda}=0. \end{align}

Now, for the case of rotations $\theta\mapsto \theta+\lambda$, our conserved quantity is $C_\theta=\partial L/\partial\dot{\theta}=mr^2/2$, the angular momentum. For a lagrangian invariant under $r\mapsto r+\lambda$, $C_r$ is the conserved linear momentum. Because in our case this isn't a symmetry, $C_r$ is not a constant of motion.

For the time symmetry we would need a generalized version of Noether's theorem but since in this case we are specifically interested in the relation between energy and time, observe that the derivative of energy is \begin{align} \frac{dH}{dt}=\frac{d}{dt}(p\dot{q}-L)= \frac{d}{dt}\left(\frac{dL}{d\dot{q}}\dot{q}-L\right)= \frac{dL}{dq}\dot{q}+\frac{dL}{d\dot{q}}\ddot{q}-\frac{dL}{dt} = -\frac{\partial L}{\partial t} \end{align} and therefore, $H$ is conserved if $L$ is explicitly time-independent.

Normally, observing cyclic coordinates in a Langrangian signals a conserved quantity. By cyclic I mean ignorable or basically not present. You should see Goldstein for this.

Secondly Noether's theorem will allow you to identify constants of the motion. To answer your question though, yes we can simply observe conserved quantities by looking at the Langrangian of the system. You may enjoy looking for conserved quantities in some relativistic space time metrics such as that of Karl Schwarzschild. This could be a fun and educational example. There are many more of course. Good luck.