Important concept: don't think any of this will be easy until after you have some experience with it, and you have wasted some materials, and you have sworn some mighty oaths. Your first attempts will not be a walk in the park, and the results may not be pretty.
Yes, the cheap laminators on ebay can probably do the job, if the particular one you get is not a lemon. The trick is learning to adjust them correctly, which is much harder to do on those cheap laminators than on a good one. For mounting 24" prints you would want about 25.5" wide adhesive, and a 29" wide roller laminator. Look for a laminator that has "feet" on the two fold down panels, because those infeed and outfeed tables will be wider and more useful than the shorter infeed and outfeed tables on the latest crop of cheap laminators without those feet.
As for hot dry mounting with drymount tissue, versus cold adhesive mounting, it all depends. Using extremely rare perfect technique, a dry mounted print on a high quality substrate will probably stay down longer than an adhesive mounted print, but variations in operator skill levels might switch that around. Drymounting is limited in the range of materials it can attach, for instance you can't drymount to PVC. For packages like the one shown in the photo, adhesive mounting would be the better choice, or at least the more practical choice. When working with adhesives, be sure you are working in a nice warm room above 75F, with all the materials at the same temperature. BTW "laminating" is when apply a clear sheet over a print, it's not mounting.
In your picture we see (I think) a photo mounted on a thin panel, on a spacer, mounted on a larger panel, sunken into a shallow floater frame.
The photo is mounted on a panel like Masonite, or Gator, or Dibond, or possibly even 3mm Sintra (aka PVC). The print may be wrapped around the edge, hard to tell. You wouldn't be able to wrap RC paper, so the closest you could come is perhaps black, 3mm PVC or 3/16 Gator. The edge of black PVC might look nicer. Since the print and its spacer support will be attached to the back-most panel, you probably wouldn't want to use something as heavy as Dibond or Masonite for the print and spacer, although you could. BTW 3mm PVC can be cleanly and relatively easily cut with a utility knife and a clamped-down straight edge, but 6mm PVC is a royal pita and weights a lot.
The print is floated on a slightly smaller spacer panel. I will guess the print and it's back panel are adhesive mounted together, and then that assembly is adhesive mounted onto the larger back panel. If the print is mounted on 3mm PVC, keep the spacer panel size close to the print panel size, to avoid flex in the unsupported parts of the print panel.
I have some opinions about what that frame should be, but I won't share them because what works for me might not work for anybody else.
The large backing panel could be anything that's nice and rigid. You'll have three large panels, each contracting and expanding to its own music, with two layers of adhesive or glue holding them together, so make sure you do everything as wonderfully as possible, and good luck.