1. Overpowering the sun requires you to have a lot of power or be very very close to the subject, generally a 360~600ws light will be needed for this to have any kind of distance between your light and the subject with a modifier and be able to beat down the sun at the worst time of day. In the morning and evening it's easier with less power. But I would suggest that this starts in the 300+ ws range, and really, 600ws is a good thing to have if you plan on doing couples with a big modifier and want to actually beat down the sun. This costs more of course, but that's just how it is. A single light to do this would be 1/3rd to 1/2 of your budget. Especially if you want it portable with a battery (internal or external). Also you'd have to consider if you want pulse HSS, hypersync or TTL options, etc.
2. YN622's can trigger anything you can plug them into via pc sync cables, so you're good for almost anything here except lights that lack pc sync ports.
3. Again this is light specific, also, not all lights are really HSS, some are just long burn and some are pulse, you have to really do research to know what matters to you and which will do what you need from it. You also should consider TTL if you're in this price range too as it's become available lightly.
4. A good light stand also costs quite a bit, 1/10th to 1/20th of your budget easy.
Also, what kind of support do you want for your lights (ie, warranty)? It may matter to you compared to just getting some lights that are basically a purchase and if they die you're stuck, versus having a store where you are (USA at least) to call, deal with, do warranty with, etc (for example, Adorama or another company, versus buying on Ebay from China, etc, to save a buck).
You also need to consider that not all lights use the same mounts for modifiers (like AlienBees, some Flashpoint, etc), as they have their own modifier mount types, which makes you buy into that mount type or to use adapters. There are more universal mount types (bowen's & umbrella type) that keep you a little more free to get modifiers that will fit more things as you change gear over time. I would also work this into your equation so you're not rebuying modifiers later on which can really crank up costs if you swap out lights later.
I think $1,000 is a bit too modest for a bullet proof setup to take to weddings (I would want a backup, on top of the this). Even some good 360 Monolights with battery packs are going to run $300~350 easy, each, and good stands to hold them will add to cost, plus modifiers. That already saturates your budget completely depending on the modifiers. It also depends on how compact you want things. And again, I would always have a backup to this for a wedding or any other serious even that cannot be re-done.
As for lights, the latest Godox stuff are good. Adorama's flashpoint line often is just one of these lights, rebranded, and supported here in the USA. I actually use several different lines of Flashpoint from Adorama from the 600ws portable battery strobes (Rovelight 600B) to their budget series strobes (which I use for indoor studio stuff to avoid batteries when doing product shots). They have some great Strobes, including the Flashpoint 360 monolight with its battery pack, which is probably the range you're looking to be in for size, portability, weight, and output.
I started out with Bowens mounts for my big modifiers, but over the years, I've migrated back to umbrella style mounting modifiers so that it can be light, simple, portable, collapsible, etc, and work with literally everything. I really like umbrellas (brolly box softbox umbrella style softboxes from Godox/Neewer; and things like the Photek Softlighter II (big and soft), and big PCB umbrellas for bounce reflection). I use them with my speedlites and even my 600ws strobes.
Here's some examples from Flashpoint (Adorama rebranded strobes):
Flashpoint Streaklight 360 - Portable HSS capable strobe, great for locations
Flashpoint M series - 600ws strobe, can add a battery pack. Lots of power. Good strobe.
Flashpoint EXPLOR - 600ws, HSS, TTL, with battery. Ultimate location strobe.
Adorama Boom (I use one similar, holds my 600ws, but mine is a Cheetah))
Otherwise, look at Cheetah too. They're also rebrands. But you go to them for the service too.