Electrical panel ground issue
We see a lot of really bad electrical work here.  This is not that.  I wouldn't lose any sleep if my panel looked like this.  There are a few defects. 

If you're really worried about electrical fires, inspect all your wiring and install AFCI breakers on all the circuits which are not in steel conduit, or which have flexible cords plugged into them. 

Ground doesn't necessarily come from the service entrance

Nothing says the ground must come in from the service entrance.  The two hots and neutral do.  Ground is derived from rods, water pipe, etc. and brought into the main panel/breaker.  

Your situation is a little unusual since you have a main breaker in a different box from the main panel (which does not have a main breaker).  It appears your ground comes into the main-breaker box.   Make sure it goes to somewhere which is an actual legal ground.

You can see where it bonds to neutral there, and then bonds to the case through a flat strap.  This is the only place ground should bond to neutral. 

Ground is carried from the main breaker to the main panel via the metal conduit.  Metal conduit is a valid grounding path. 

Is neutral tied to ground in the main panel?

The main panel has all its neutrals neatly separated from all its grounds.  That suggests good work.  However, I am concerned that it appears there's a bond between the neutral bar and the ground bar, and I don't entirely see how the ground bar is bonded to the actual metal case of the breaker panel.  There should not be a neutral-ground bond in more than one place. 

Fill the empty holes in the panel

The knockout holes should be capped. They make plugs just for that.  But first... 

Too many conductors in that conduit

Look just to the right of the main cable entrance, you see a conduit with 5 hots and 5 neutrals going into it.  If the conduit is longer than 24", that's too many - the maximum is 4 circuits per conduit**.

So, lay a second parallel conduit and move 2 of the circuits to it. Don't pick any 4 random wires, always keep hot together with its partner neutral.  

If you must put 5-10 circuits in a conduit, you must de-rate, meaning use thicker wire: 12 AWG for a 15A circuit, and 10 AWG for a 20A circuit.  



** The gory details are in NEC 310.15b3a. You are allowed up to 9 conductors in a single conduit, regardless of size.  Grounds don't count.  In a 240V circuit, neutrals don't count since they only carry imbalance current.  

Above 9 wires, you have to "de-rate" wire capacity.  For 10-20 wires, use 12AWG for a 15A circuit or 10 AWG for a 20A circuit.  For 21-40 wires, use 10 AWG for 15A, or 8 AWG for 20A.