I've always used upholstery thread and needles. Sometimes you need an awl too, especially if you're making something instead of repairing it. Sometimes I'll wax it with carnuba wax to keep it from binding. If it's really thick, like a handle for a press camera that I'm making from multiple layers, I'll drill out the holes first. You're not supposed to, because it won't grip the thread well, but sometimes it's your best option. If you use a bit smaller than your thread's diameter, it'll still grip. I'll use contact cement to hold things in place before punching the holes and sewing. The type of stitching used is determined by the type of connection you're making. Look up shoe repair and saddle repair videos for more info on how to do proper leather stitches. They're different than traditional fabric stitches. Also, if the leather is stiff, apply some conditioner to it and let that absorb into the fabric. It'll make sewing easier, and will leases the likelihood of ripping the leather. If the leather is dry enough, it might take a few applications. If it's too far gone, you might be best off replacing it. Leather can only be brought back so far. At some point, it's not worth working on, as you'll spend hours repairing something that will just fall apart again.
I'm not claiming I do it correctly, but I've been repairing or making about one or two leather items a year ever since I was a kid in Boy Scouts, and never run into issues with my techniques.
In my experience, when the stitching has gone the leather is in pretty bad shape. I have though of getting thin strips of leather to serve as reinforcement and gluing it all together, but have never trying it. So many old cases, so little time...
I've done this too. It works. Use contact cement to hold them. However, it's a bit of a pain and doesn't work well if you have to repair large areas. Thin leather doesn't provide much support and thick leather gets in the way, as usually these cases don't have much room. It's good for a small spot, but if you have to do an entire seam, I think you're better off just replacing the entire piece of leather. Use the old one for a pattern and then dye the whole piece darker to match.