I use a Sigma 24mm Art and it's one of the very best lenses I've ever purchased. On your D7200 it'll give you an effective focal length of 36mm, but this will be fine in practice.
Your other alternative in the Sigma line up is the 20mm Art. This lens is similar in price, though it's heavier and it has a permanently attached lens hood. I personally wouldn't recommend this lens as it lacks a filter thread, unlike it's 24mm counterpart.
While I'm on the subject of filters, I'd strongly suggest you get a polarising filter to use with your landscape photography. This is the polarising filter I use with my Sigma 24mm, and it's especially useful in bright/hazy conditions.
Your Nikon D7200 is an excellent camera, I've come across several pros who favoured it over their D4s for it's crop factor and lightweight build. I wouldn't look to replace it just yet, unless there's something in particular you don't like about it.
I hope you found this info useful, let me know if you want me to elaborate on anything further. Cheers for now.
the 24mm is exactly the one I've been considering, along with polarizing and neutral density filters, so I'm glad to hear you love it - I've seen nothing but praise. I'm trying to cut down my gear bag as much as possible. I climb/hike with a Cotton Carrier front harness system and a three lens changer at my belt so I can carry whatever I need for living on my back; the reasons you touch on are why I've stayed with the 7200 so long.
However, I was shooting an art gala event last night, and I found on continuous high, given the complex lighting and movement in the space, it was bogging down worse than normal, even if I was only taking three or four photos in quick succession. I think if I continue with special event/concert photography, I may need something that can think a bit faster on the fly.
If weight is major concern, you might also consider the Tamron 24-70mm 2.8. It has excellent sharpness, and with the crop factor it could replace several lenses in your bag. I even knew a guy who very successfully filmed a documentary with it. He swore that he wouldn't mind if someone welded the lens to his camera, as he's never taken it off since purchasing it.
As for your D7200, I'm not sure if you were referring to the shooting or autofocusing speed, as the lighting wouldn't affect the frames per second. If you go into our camera's settings menu, under "Autofocus" you should see two separate entries labelled AF-C/AF-S Priority Selection (probably under a1. & a2. respectively). Set these both to "Release" instead of "Focus". This will ensure that the shutter will continue to fire even if the camera doesn't believe it has attained focus.
You're probably already aware, but a faster memory card can also help with stuttering issues. Another trick you might try is changing the RAW bit depth to 12-bit lossless compressed. This will help ensure that the frame buffer doesn't fill up so fast, and will allow you to shoot continuously for longer.
I'm not sure how much of this you are already familiar with, but let me know if you have any questions.