You may check the status of your plan check either through 311 or via Department's online permit and inspection report. You can reach this online service at Online Services. 

An inspection card is not required for any Electrical or Mechanical permits.

Yes, if the installation involves a (1) communication circuit or class 2 signaling circuit over 30 volts or under 30 volts but more than 100 va, (2) amplifier circuits not using class 2 or 3 wiring, or (3) any circuit operating over 15 volts that control, generate, transmit, transform, or utilize more than 25 watts or va. See section 93.0202 for more details.

A product (or equipment) is considered approved if it bares either the approval label of the City of Los Angeles Electrical Test Lab, or listing or certification label, symbol or other identifying mark of a recognized testing agency. And in either case, the product (or the equipment) is installed and used according to any limitations or restrictions posed by the testing agencies and the requirements of the code. See section 93.0402 for more details.

You will need a plan check when the scope of work involves items under code section 93.0206.

Any time you alter, repair, add, relocate, or modify existing electrical wiring, you will need an electrical permit. Refer to section 93.0201 for a list of exceptions.

If you have resided in the house for 12 months or more.

A contractor may sign for their own design as permitted by the State Law and section 93.0206(a).