Should the lambda for L1 norm regularizer inversely be proportional to the number of trainable weights?
Backpropagation doesn't handle Regularization like this i.e. if you are thinking "10 weights make penalty 100, so 100 weights will make penalty 1000. 
So let's have a smaller $\lambda$

Backpropagation uses the partial differentiation of the Loss. Now the Loss has an extra $\Sigma$$w_i^2$ and so the derivative will have an extra piece which will be proportional to the weight(Derivative of $x^2$ = $2*x$). 
It will be separate for every weight, so the number of weights will have no impact

The new equation will be - 
$w_i$ = $w_i$ - (backprop stuff as before) - $\lambda$*$w_i$

$\lambda$ is just a knob. You also have any value proportional to $w_i$ e.g. 5% of $w$.