Do you "need" a condensate trap - YES, unless your unit specifically comes equipped with an internal trap, you need one. Many package and split systems require the installation of a dedicated condensate disposal system, including a trap. Check your installation documentation. Bryants do have in some cases an internal trap.
2.) Either way, a trap is more than a drain. I quote here typical condensate trap information from a typical well known package unit manufacturer's install documentation.
"Do not operate unit without a drain trap. The Condensate drain is on the negative pressure side of the blower; thererfore, air being pulled through the condensate line will prevent positive drainage without a proper trap".
Yes, the bolds in this quote are from the manufacturer, not me.
Once you install the trap you must PRIME it to get it started. Do not depend on the condensate from the unit to prime it. Every minute your unit runs without a trap, or with an un-primed trap it is sucking warm outside air, dirt, bugs, mold, etc. right through the trap and right into your conditioned air space, because that entire circuit is "ON THE NEGATIVE PRESSURE SIDE OF THE BLOWER".
Most traps need about 3" lift to work correctly and allow for static pressure fluctuations.
Besides the dirt/mold/warm air/fiasco, your current situation is draining condensate onto the chassis of your unit. This will lead to corrosion - which you can already see forming a little in the picture, because condensate is water, and can also be slightly acidic.
Finally, did you not notice that your installer left your unit on the delivery skid when he installed it? Was this a deliberate choice or simply an oversight? (there might be reasons why it was left on the skid, I suppose, but it looks tacky)