"Should the secondary line leak into the bathroom constantly?"

No.

"How can I fix it on my own without paying $300+ I’m getting quoted for an HVAC repair?"

I would consider $300 cheap to fix that and sign up in a heartbeat. Unfortunately some things are tough to DIY.

I am not sure I am following how this condensate drain works. Is the piping with the trap (pipe bend) supposed to be the primary drain? Is so where does the water from that line drain into? Is the second condensate line (the one towards the back) the one that drips into the bathtub?

If the line with the trap is the primary and drains into a plumbing drain, then I think the problem is the trap is clogged and all the water is draining into your bathtub. The drain into the bathtub is meant to be the emergency drain in case the primary drain fails. The day a drop of water comes out of the ceiling drain is the day the primary drain has to be fixed.

I am not sure why there is a trap in the primary line. It seems to serve no purpose other than collect debris and cause a clog. But if a is needed I would install a clear EZ-trap. This would allow you to see if it is clogged and open it up to clean it.

Yes, the primary line is the one with the trap and it drains directly into a plumbing line (sinks) that are directly under it. The second is the one at the back with no bend and it’s leading directly over to the bathtub.

I considered taking out the trap section and replacing it with a straight piece as the line makes a slight angle downwards and let gravity do the work, but I hadn’t considered replacing it with a clear EZ-trap where I can see a clog - Thank you!

If the primary is connected to a plumbing drain then you should have a trap to stop sewer gas. If there is a trap further down then you don't need the trap by the air handler. If there is no other trap, then the one in the picture is not doing anything, The sewer gas can escape from the open clean out pipe.

Here is an EZ-Trap example. It even comes with its own brush for cleaning. If you install this you don't need to open clean out pipe. All that does is collect debris.

OP, one thing missing in the image I see is that the equipment assembly that includes the coil isn't sitting in a pan, as with water heaters. And probably should be, though it may not be possible to slip one in under existing equipment. The pan would itself have a separate pipe run to drain outdoors or to a different part of the house's outflow system. And, often a water leak monitor to sound an alert should water be accumulating in it.

I agree that you should call a HVAC tech, the line would seem to be clogged.

Where I am traps are not required. I have one at one place (equipment in an attic) and not at another with two systems (equipment not in an attic). Both new systems I had installed in the past few years. The attic equipment drains to an under sink connection with the secondary on the underside of a masonry window sill outside. The other place without traps, the condensate pipes drain alongside the foundation.

For the undersink drain connection, a piece like that pictured below is part of the vertical sink drain assembly. The connection itself slopes upward and is many inches (6?) above water level of the sink trap.

it’s rusted the metal and turned the tub brown.

That is probably an indication that the drip pan has rusted and needs to be inspected and probably replaced.