Is outlet box required to be mounted

Read inside the box. If you install 14 AWG conductors to the new box, instead of 12 AWG conductors. You'd need at least a 22 cu.in. Box, or a 3" x 2" x 3 1/2" device box.

• 8 for current carrying conductors.
• 1 for grounding conductors.
• 2 for device fill.

11 total, times 2.00 cu.in. for 14 AWG conductors = 22 cu.in.

However, since you're using 12 AWG, the calculation looks like this...

• 6 for 14 AWG conductors x 2.00 cu.in. (12 cu.in).
• 2 for 12 AWG conductors x 2.25 cu.in. (4.5 cu.in).
• 1 for grounding conductors x 2.25 cu. in. (2.25 cu.in.).
• 2 for device fill x 2.25 cu.in. (4.5 cu.in.).

Total = 23.25 cu.in.

For more information on box fill, check out this answer

As for connecting the new wire. There should be no problem attaching it to the receptacle. However, since you're using 12 AWG conductors, you'll have to use the screw terminals.

If you want to do pigtails, you can. I know some guys don't like the backstab connections, so if you want to avoid using them you can do pigtails. Ideal 76B® Red Wire-nut® Wire Connectors are rated for 2 to 5 #14 conductors, but check with the manufacturer of the connectors you're using. Again, if you're going to do this, you'll want to use #14 conductors instead of the #12 you've installed.

Sounds like your assumption that water penetrating the box is correct no need to figure out any more than that. These boxes are not made for direct burial. (Pretty amusing actually. I will be using your picture as a teaching tool.)

The conduit and wiring the box is attached to needs to be extended upward I would say a minimum of 12" above grade. It also needs to be attached to something to support it. It cannot "free-float". Perhaps you could attach it to that pressure treated post in the corner of your picture.

Extending the conduit, if it is PVC, is just a matter of gluing more conduit on to the existing. However, extending the wiring will involve pulling new wire from the last existing junction box or the panel. Wire cannot be spliced inside the conduit.

If these lights are temporary, such as holiday lighting, they can be cord connected. The receptacle needs to be GFCI protected either by a GFCI breaker or receptacle. Also, the receptacle needs to be weather resistant and tamper resistant. Finally the cover needs to be an "extra duty, in-use" cover to protect the connections while the cords are plugged in.

If these lights are permanent, (and it sounds like they are: landscape lighting and accent lighting) then a permanent wiring method should be used instead of cords. So, you could still extend the receptacle for a convenience receptacle, but you should hard-wire all the outdoor lighting.

Good,luck!