Yea you'll do just fine.
I always carry a Styrofoam box in the car *just in case* I find fish while out & about. The nice thing about them is not only do they keep fish warm, but my biggest plus on using them is its dark - Fish are stressed less. Any home store will have you different sized buckets with lit's, just get any dark color for them if you're buying new one's anyway. You CAN gladly get a 5g...me I use the smaller one's cause I'm just a tiny person with limited arm strength
Alright here's how I'd bag them guys:
Betta sep in his own bag
White Skirts & Rasbora in 1 bucket
Corys in 1 bucket
ADFs in 1 bucket
So you'd need 3 small buckets (~2-3g) reason I sep Betta is obvious. ADFs can get upset when stressed, so just to make sure they don't stress the Cory I'd sep them. I know its sounding stupid to bag 2 fish alone, but rather safe then sorry right?
Naw just like I said in order to catch them all in the frist place take out your larger plants, these gotta go in some water anyway so just stick a few plants in to each bucket and that'll do the trick right there
So dep on how many big plants you have there you should be either able to spread them among the 3 buckets or need a 4th one in which you can then also toss your filter pads, wheels etc. (PS see if you got a Lows, I recently got some there with lids for $2 each in dark blue they were like 2-2.5g)
And I'd NOT keep old tank water, its full of waste and they only thing you'd do is carry over waste and that's not necessary nor beneficial.
Just fill up the tanks in the new house with water conditioner (prime etc) and get the heat & filters going.
And jjuusssttt to be safe, have a working liquid test kit on hand to check the water over the next few days, nothing should pop up....but we all know tanks and nothing rarely goes by the books right
And for your "harigrass" tank, make sure to leave 1-2" water in it just so the plants stay wet and don't dry out on you.