p.1 #2 · Monopod vs Tripod...which use more? $500-700 range.
Ok, I'll bite. First off, you can't assume that support will fix your softness. Many other factors can be an issue.
Obviously a tripod is more stable than a monopod, but it is more cumbersome to setup too. I carry one or the other most of the time. I use a monopod when just a tad of extra support is adequate, mostly with long lenses and when running-n-gunning at weddings and events. A monopod acts as a substitute for lens image stabilization.
I use the tripod when I need real stability and/or careful composition. With a tripod, I can easily do exposures of 1/2 to several seconds. You can't do that with a monopod.
You'll be hard pressed to spend $500-700 on a monopod, even with a RRS head. On the flip side, you could easily spend more than that on a tripod. I won't recommend brands, but a tripod is a durable product that you are better off spending extra on. The tripod will be useful long after the bill has been paid.
p.1 #4 · Monopod vs Tripod...which use more? $500-700 range.
Go with a monopod and pistol grip ball head. It sounds like this would be what you need according to your profile. I think a tripod would be overkill in your situation, figuring you use mostly wide angle and short teles. Add a light travel tripod for night work.
p.1 #5 · Monopod vs Tripod...which use more? $500-700 range.
you could certainly get both with that kind of a budget. Though as mentioned above, tripod's can get much more expensive, I have about $1600 in mine and still find it lacking at times. Tripods are always a trade-off in my opinion, Cheap (relatively), sturdy, and light. Choose two.
p.1 #6 · Monopod vs Tripod...which use more? $500-700 range.
Why not try a $50 Monopod, like the excellent Bogen 680B or similar first and see if it might work for you before you go spend $500-700 on one.
Heck, you can pick up a cheapie at Walmart for $10 or so and get a far better idea of what would work for YOUR needs. Maybe you'll find you hate carrying one or you prefer working from a tripod.
Unless of course you like dropping $500+ on what a bunch of strangers on the internet tell you to do with your money
p.1 #7 · Monopod vs Tripod...which use more? $500-700 range.
I love my monopod, it has other uses such as a walking stick while getting to the perfect spot through mud and snow….I leave the strap around my neck and let it lean away from me when using my bag or getting into my pockets. It's just so versatile. Giottos rock btw!
p.1 #8 · Monopod vs Tripod...which use more? $500-700 range.
A tripod is going to be more stable overall - assuming it's a sufficiently rigid tripod/head to begin with. Tripods can be more cumbersome to deploy and pretty much always take up more space. There are places that might allow use of monopods that don't allow tripods.
I have a lighter tripod that I use for city walks and other occasions where I'm not expecting to use heavier lenses nor is it terribly wind resistant. It does need to be used with some care in the "mirror slap" range, sub 1 second or so range, lock-up, remote, etc. It's not so heavy that I can't use it as a monopod and under some circumstances, I've been able to use it legs together braced against a solid support like a wall, guardrail or sturdy fence, etc., when there wasn't space or time to deploy the legs. A bungee cord or two can allow you to strap in a collapsed tripod or monopod to a solid object as well.
I also have a somewhat collapsible walking stick (too long for carry-on) that can mount a camera or head and use that at times. So I haven't felt the need to get a photographic monopod although I had one years ago.
Because I do sometimes strap down or brace the pod, I do find a ballhead handy and the pistol grip head particularly useful on the walking stick. However, if not braced down, etc. a "loose" ballhead isn't real handy on a monopod as compared to the same use on a tripod.