I've been going back and forth in your photos and I think I grok it.
Somebody fouled up the installation.
You show the supply from the house as you have labelled in yellow - but those wires cut off in the larger photo.
Those wires are feeding
into the
exit of the breakers on the bottom.
They should feed the supply into the bottom of the box in those heavy hex set screw lugs, cross to the right in teh piano key pattern and towards the right, through the breakers and out the wires on the right side.
Switch off the power from the breaker in the house panel
run an extension cord and lamp from the house to the garage at the panel- working in the dark sucks
dis connect the supply wires coming into the garage subpanel from the screws of the breaker, this is the wrong installation.
Connect them into the lugs I've circled in this photo
Which side is black or red doesn't matter
That frees up that bottom 220 30A breaker doesn't it ?
Pull that breaker - it prys / snaps out.
Replace it with the 20A
Possibly this one
http://www.homedepot.ca/product/double-pole-15-amp-qo-plug-on-circuit-breaker/901406
Take your 30A into the store with you - you know it fits your panel
(they have trouble with theft and keep them under lock and key, put a name tag on your 30 so they know it's yours
They also are finicky with returns on breakers, eyeball your 30 that you know fits the panel side by side with the 20A you are buying
be sure that size and lug positions match so it will fit and snap into the panel before you pay for it.
Check and see if the QO model matches too.
http://www.chapters.indigo.ca/books...-province-P-S-Knight/9780920312384-item.html?
Install it
Be sure there is only one wire per screw exiting each breaker.
Now to wire the outlet
Code says, plastic coated wire inside the walls where it is protected is OK.
Armour coated (Romex or metal spiral wrapped BX ) outside the walls where it could be damaged
Get the armour coated and fasten to your wall for simple installation.
http://www.homedepot.ca/product/14-3-ac-90-armour-cable-20m/908348
If you are doing 15A this 14 gauge should do it, buy it by the foot not the whole roll.
The outlet
http://www.homehardware.ca/en/rec/i...acle/_/N-ntk2a/Ne-lrv1/No-48/R-I3665304?Num=0
and a simple box,
http://www.homedepot.ca/product/utility-box-2-3-8-in-wide-1-7-8-in/977520
and the cover with the single round hole that fits
http://www.homedepot.ca/product/utility-15a-single-receptacle-cover/977523
and 2x 1/2"npt clamps - can't find a link
nuts
http://www.homedepot.ca/product/1-2-in-locknut-bag-of-10/910002
That bad installation should be a clue to get the house and garage double checked to see what else they have done you didn't find yet.
For instance, that white wire shouldn't be there coming out of the breaker.
http://www.bladeforums.com/forums/attachment.php?attachmentid=310143&d=1351734691
Verify that the supply wire from the house to the garage sub panel is 10 gauge, if they went cheap and thin you would be overloaded.
10 gauge should be .102"
Code used to require some drywall behind the panel to act as fire insulation and protect the OSB plywood from burning if the panel overheats a little.
Check and see if that still applies in your area and add it to your garage subpanel.