First off, I'd like to thank those of you that saw the title of this thread and still took the time to click it. Brave and helpful souls, the lot of you.
Much like an annoying ex, I'm coming back here because I've got a problem that only you can solve.
Now, a bit of context. I purchased the car in April of 2018, so relative to how long it's been around I'm still a fairly new owner. Had a GMC Sierra prior, but didn't have to do much work with that vehicle. Colour me inexperienced.
'98 Pontiac Firebird T/A, 5.7L | 4L60E (inb4 'Well ur problem is having a 4l60E' )

So the first thing's first, I'm going to list off everything that's happened as of late, then I'll list what I've done to solve those problems.

1. Code P0758 - 2-3 Shift Solenoid B Circuit Malfunction - Car acts as if it's in limp mode, only third gear, manual second, and reverse.
2. Left turn signal is now solid. Driver's side daytime runner only comes on when parking lights are turned on.
3. Aftermarket radio is noticeably quieter now. Volumes that would've blown my ears out before are almost required to really hear anything over the car itself.
4. Bank 5 coil-pack is causing a misfire.

Things I've done so far to fix the problems:

1. Replaced all four transmission solenoids.
2. Swapped the trans fluid (was brown when swapped, yikes.)
3. Cleared the codes, flashed the PCM, had a mechanic hook it up to a Tech 2 - all solenoids working and the transmission is 'shifting' into 1st according to the scanner, but it's definitely staying in 3rd.
4. Replaced the bulbs for most of the lights on the driver's side.

Closing thoughts:
Send help! - Seriously though, I've been mulling about the possibilities. A gentleman from a local trans shop said my trans problems could be wiring, but for it to still 'shift' and for all four solenoids to work fine I'd imagine my wiring's fine. I could be entirely wrong. I've looked into it potentially being my PCM (ECM? TCU?), my wiring harness, or even this faulty coil-pack. Another thing I've read on various threads is that I should look at my ignition switches.

I think before too long I'm going to try and replace this coil-pack to see if that'll resolve anything for me.
Thanks again for taking the time to look at this thread, and moreso for any answers you might have regarding my problems.

 

Check grounds wires, a bad ground can wreak havoc on a car.

 

The "quiet" radio might simply be due to a speaker or two which has failed, the stock speakers are known for this (especially if they have seen some aggressive use over the years.) Or, the amp may have failed. Either way, with an aftermarket head unit in place, you really don't know for sure what other components have been changed over the years - so no telling what might or might not still be original.

The transmission might have other issues beyond just the solenoids which were replaced. The brown fluid suggests, at the very least, a lack of maintenance over its lifespan. There may be other failed/failing parts within.

I agree with getting that misfire on cylinder #5 addressed before going any further. Are you sure it's the coil pack though? You could swap it with another and see if the misfire moves to a new cylinder. There could be other issues causing a misfire on #5.

 

Since it hasn't been answered yet... your front lamp problem is because of a burned out bulb, a corroded socket, or a broken wire between the DRL module and the lamp (in that order of probability). The front lamps use dual filament 3157 bulbs with a dim filament for parking lights and a bright filament for DRLs and turn signals. One filament can burn out while the other is still fine. That's why you see the lamp light up when the parking lights are turned on - it's a different filament than the DRL. When a bulb is burned out on one side, there isn't enough current to make the flasher flash so the remaining lights on that side just go on solid. You can prove it by turning on your hazard flashers and walking around the car looking for the one that is not flashing (in this case the left front).

Now these cars are also infamous for getting water in the front lamp housings causing serious corrosion of the bulb socket. This can cause the same symptoms because the bulb isn't getting a good connection. Depending on how bad it is, the socket can be cleaned or it might need to be replaced... replacement sockets are a few dollars at any auto parts store.

Finally, if the bulb and socket are okay then you may have a break in the blue wire somewhere between the DRL module (behind the radio) and the lamp. The two bulb filaments are on separate circuits - blue wire for turn signal and DRL, and brown wire for parking lights - so you can still have one work while the other doesn't. This is actually quite unlikely unless you've been doing some work that might have disturbed the original wires.

There are even more remote possibilities such as a bad turn signal switch but I would check the other possibilities first.

 

The socket for the DRLs are notorious for burning out in these cars. You can get new sockets with wiring pig tail to replace them if they look burned. They get hot, it is a design flaw. I have replaced mine twice. As already mentioned by WhiteBird00, it is a duel filament bulb. Dim filament is for the park lights, bright one is lit when daytime running lights are on or turn signal is activated. Should be simple to determine if it is the bulb or the socket. If you replace the sockets, use wiring connectors that heat shrink the plastic ends, makes it more waterproof.

 

Along the same lines as above the turn signal flasher can go bad if the socket is bad

Turn on the hazards to see if they work, if the lamp doesnt work then it's a bulb / socket issue. If the hazards work then the flasher is bad