If you look at a Nikon, Canon, of Sony teleconverter, you will see that the teleconverter protrudes into the lens beyond the mounting surface. Many lenses do no have space to accommodate this protrusion--the teleconverter will damage the rear lens element.

If your lens can accommodate a TC, you may not be able to autofocus if your lens maximum aperture is too small. Your AF system requires a sufficient light to be able to focus quickly and accurately. That amount depends on the AF sensor in your camera. When focusing, your camera will change the lens aperture to its widest setting (i.e., f/4, f/2.8, etc.) to maximize the light available to the AF sensor. When insufficient light is available to the AF sensor, AF will slow down and only the more central AF points may receive sufficient light to focus. Your camera manual should tell you the autofocus limitations with respect to the lens maximum aperture. For example: for maximum aperture f/2.8 to f/4.0, all AF points are available; from f/4.4 to f/5.6, a central set of AF points are available, and from f/5.6 to f/8 only the center AF point is available. (The specifics differ by camera. Beyond f/8, most cameras will not reliably autofocus.)

When you add a teleconverter, the optics of the TC narrow the field of view (appearing to you as magnification) and, reduct the amount of light from the lens that falls on the image sensor. This effectively reduces the equivalent maximum aperture of your lens. You effectively reduce the maximum aperture by one stop for a 1.4TC and two stops for a 2.0TC. So your f/4 lens mounted on your camera with a 1.4TC becomes f/5.6; with a 2.0TC, f/8. If you have a zoom lens that has a maximum aperture of f/5.6-f/8.0 (unzoomed-zoomed), with the 1.4TC it becomes f/8.0-f/11.

For focusing with small aperture lenses (f/5.6 and above), a brightly lit target and minimizing camera motion become important for autofocusing.

See: https://photographylife.com/product/sensor-gel-stick-for-sony
I use this gel stick. This is the only gel stick you can safely use on Sony image sensors.

Get the Canon 18-135. Most of your shot opportunities will be between 18 and 35, but the longer focal length will occasionally be useful.

No significant difference.

An Olympus E-M1 Mk 2 with a 100-400mm or 300mm lens will weigh 3.4 or 4 lbs respectively (EFOVs of 200-800mm and 600mm). A Fuji X-T2 with a 400mm lens will weigh 4 lb (EFOV of 600mm).

Probably a software problem, not a hardware problem. I don't have a problem running Lr and Ps on a Mac with a 256 GB SSD, i5 processor, and 8GB of RAM. Of course they would run faster with a faster processor and more RAM. (Adobe recommends at least 4GB of RAM.) After you fix your software problem, if your computer is too slow, look into increasing your computer's RAM and upgrading to a solid state drive (SSD).

I hope you are joking!

I'm sorry. I don't quite know how to explain this to you. But I'll try. Forgive me if you were really joking.

Your camera records luminance (amount of light). It uses filters to separate the light into red, green and blue channels and records that data in a RAW file. ALL colors you see in each pixel of your image are calculated from the luminance data in each of those channels for that pixel. When converting to B&W, your software is using the luminance data from each of those channels. Your B&W image is, in fact, black, white, and all of the shades of gray in between black and white. Lost information results in abrupt changes in tone (level of gray) and/or some tones being combined into a single tone.

Also note that each time you open, change (no matter how slightly), and save a JPEG file, it is compressed again and loses even more information.

Some people are more perceptive than others. If you process a JPEG and a RAW file to B&W and can't tell the difference, that doesn't mean there is no difference. It just means you aren't as perceptive as most of us.

Banding is due to 60 cycle/sec florescent lighting and causes banding in the red channel at some shutter speeds. Why would you be shooting sports indoors at 1/60 sec? I expect an update from Sony to fix this.

GX85 for best image stability.

The quality of your image is dependent on the availability of information, not the colors. You have more flexibility in processing RAW files to B&W. JPEG files are lossy compressed RAW files--information has been lost permanently--an inferior starting point for processing.

I would use whatever is mounted on the camera.

Consider:
Nikon D810 with 200-500mm f/5.6: 7 lb (I have this combination)
Nikon D810 with Tamron 150-600mm f/5-6.3: 6.25 lb (I have this combination)
Nikon D810 with 300mm f/4 pf + 1.4TC (f/5.6): 4 lb (I have this combination) Effective focal length = 420mm.
Nikon D810 with 300mm f/4 pf + 2.0TC (f/8): 4.25 lb Effective focal length = 600mm.

The 1.4TC works on the 200-500mm as well as the 300mm, but the 300mm+1.4TC is a sharper combination. I haven't used the 2.0 TC, but I imagine that sharpness will suffer.

I actually use the 200-500mm, the 150-600mm and the 300mm pf on my D500 for sports and wildlife. The 200-500mm feels much heavier to carry and tires me out shooting handheld where the 300mm pf lens is much easier. On the D500, the 300mm pf + 1.4TC has an effective focal length of 640mm and weighs only 3.8 lb. That is what I take to shoot birds in flight or on hikes.

You may lose AF with the D300 at f/8. If light is good, you may be able to focus with the center AF point--slowly and sometimes with some difficulty.

What computer; how old; how much RAM; how big hard drive; what photo application? Your problem may not be your computer.

Rent a Sony A7S2 with a 24mm, 25mm, or 28mm lens. Whatever camera/lens you use, test it in a dark room to see how to optimize performance. You may find it difficult to autofocus--good to know and figure out a work-around before you go.

Mondo, no one can answer this question for you, because they don't know what will be acceptable to you, they don't know what typical background and light is in your photos, and they don't know how capable you are in reducing noise during post processing. And they might not be using your camera and lens.

However, this is easy to determine: With the camera on a tripod and focused on a target typical of the birds you want to shoot (you don't need the bird to focus on--you can use a tree limb; the brightness of the target should also be typical) 1) set your camera to manual and auto ISO, 2) set exposure time so that ISO is 100, 3) take a series of shots reducing the exposure time by half for each successive shot (example: 1/125, 1/500, 1/1000, 1/2000, etc.). Compare the images using Lightroom or some other post processing software, after applying noise reduction (as desired), and using a zoom level consistent with the size at which you might want to print. Pick the highest ISO at which you find the noise acceptable.