Running 240v power to detached garage for electric vehicle
First, using any sort of NM cable outside is right out of the question.  Cannot do it; it's not wet-rated.  If you want to direct-bury cable, use UF or other types; however you want to run in conduit.  Running cable in conduit isn't even crazy, for reasons let's leave to another Q/A. So let's armwave that you will be using individual wires.  Fortunately those are almost all dual-rated dry/wet (THHN/THWN-2, the W being Wet).

Conduit

For this reason, it's important that the conduit is either a) continuous from the panel to the destination; or b) stops at a large junction box to make a splice from THWN-2 to cable.  

When laying conduit, keep your bends to an absolute minimum between access points; use conduit bodies at corners to give easy access. You see a conduit body in your picture, but you can't use that one because service wires are in it. 

If you are willing to trade "being able to trench with a garden trowel" for "murderous price per foot", consider Rigid or IMC conduit, which only needs 6" of cover over top of it (12" cover for vehicle pathways).  But figure like $4/foot for the stuff, and a trip or two to the hardware store to have pipe threaded.  Upside: it's a viable ground path, so one less wire. 

For PVC conduit, you need ot use Schedule 80 and you must have 18" of cover over top of it, so trench to 20". 

Wires

I'm going to list the wires by price per 100ft, since your distance is about that.  I'm getting these prices from wiresandcablesyourway.com.  All or THWN-2 or XHHW. 

 1 x #8 bare copper ground $37
 3 x #6 copper conductor   $57 x 3, 60A max  OR...
 3 x #4 aluminum conductor $26 x 3, 60A max  OR...
 3 x #2 aluminum conductor $35 x 3, 80A max  OR...
 3 x #1 aluminum conductor $52 x 3, 100A max.  
 2 x #10 copper conductor, $30 x 2, 20A


Aside from aluminum being less than half the price, one reason to use it is that puts your wire size larger than #6.  On #6 or smaller wires, you must use an actually-white wire for neutral, which can make buying wires more expensive.  #4 and up, you can mark it with tape.  Grab a 5-pack of red/white/green/blue/yellow electrical tape for $4. 

The 20A/120V circuit is a real problem. Because of the distance, you need to upsize the wires to #10 or you'll have alarming voltage drop.  Now let's do the math on this.  The two conductors are $60.  You won't need it if you get a subpanel.  A perfectly respectable 16 space subpanel can be had for $70.  That's a no-brainer if you ask me. 


With #6 feeder, you can comfortably use 1" conduit.  
With #4 feeder you can only use 1" conduit if you omit the 20A circit, and it'll be a hard pull. 
With #1 feeder you can only use 1-1/4" conduit if you omit the 20A circuit, and it'll be a hard pull. 


The ground wire, presumably #8 copper, can be shared between the feeder and the 20A circuit.  

The subpanel

Here we are huge proponents of huge subpanels.  When buying the panel, the cost of extra spaces is super cheap.  After you've fit it, and regret getting too small a panel, it's super hard and expensive.  Better to skip a pizza and get a bigger panel: 16 space, even 24 space.   You'll thank us later. 

"Circuits" are not "spaces".  A panel that claims "16 circuits" means an 8-space panel using duplex/"double-stuff" breakers.  That's deceptive advertising because most circuits these days needs AFCI/GFCI breakers, and those are not available in double-stuff.  

Other stuff e.g.  Your diagram

The weird cable with male ends on both ends is probably a "suicide cord" for running a generator. Or should I say a "murder cord" because his plan was to plug one end into a generator and the other end into the garage feed, to backfeed the garage then backfeed the house. And maybe he turns off his main breaker first and doesn't kill a lineman.  (Because when a 120V generator backfeeds a 120/240-9600V poletop transformer, the voltage on the backside of the transformer is, guess what! 9600 volts.  That's how transformers work.)  

Even worse, it sounds like rather than running a 120/240V /3 cable, he was using two separate cables to backfeed each leg separately.  What a hokey-dokey rig.  This is what fools like that do.  Just remove the plug ends at one end so nobody can plug it in. 

On the remaining buried cables, you might dig 'em up for a closer look.  If they are NM, tear 'em out - they are ruined.  NM is not wet-rated. If they are UF, see how deep they're buried.  If it's &lt;24" it's too shallow, tear it out.  Otherwise you can disconnect it, pull it through the building, slip a Sched. 80 PVC conduit over it and rebury it (at the bottom, an elbow of course), and then provide physical protection up to a proper Code building entrance.  And then put them back in service.  Free wire :)