Is the output of this LC circuit a DC current?
I'll give a general overview.

1) The output voltage should be ideally constant if you provide perfect filtering (which isn't realistic). Think of \$L_1\$ and \$EC_2\$ as a low pass \$LC\$ filter, that what it is.

2) If your input is a \$120V_{ac}\$ voltage signal, then, after some cycles \$EC_1\$ will get charged up to the peak  value of that signal \$V_{peak}=120V(\sqrt2)\approx170V \$. Minus the drop across the diode.

3) Yes. A relatively large inductor keeps current constant (e.g choke inductor) and a relatively large capacitor behaves as a voltage source.

4) Don't think of \$EC_2\$ separately. Along with \$L_1\$, it forms a low pass filter which reduces the ripple on the output signal.