The leather for the handle was likely originally hand stitched, maybe similar to baseball stitching, using a curved needle. Your local upholsterer may have a small piece of leather, if you'd like to try that sewing. If so, get 2 or 3 pieces of leather, for practice or learning from mistakes. The leather industry uses Barge cement, specific for leather. You may be hard pressed to find it in a local store. A local shoe repair shop will have an appropriate adhesive. Shoe repairman might can repair the handle.... or a saddle maker.
A pool table (w/ parts, supplies) dealer/outlet/sports center will have some thin felt for the drawers' interior lining. *Green is a nice color for tool chests. Supplier likely has the adhesive, also. Maybe, maybe... maybe(!!!) your local fabric store has thin felt for those drawers, but I've never seen as high quality of felt at fabric stores, as the quality of good pool table felt is.
If you use a spray adhesive, for felt, be very careful where you spray.... you don't want to get any adhesive on other areas. Local auto parts stores have spray adhesive, for headliner installation and also used for foam-to-foam or fabric-to-foam adhesion. As an upholsterer, I use it often and at times for similar tasks as cloth to wood, but your drawer spaces are too small for good results with a spray. This stuff is very sticky, cleans up with mineral spirits, BUT again, the spray pattern is hard to control, so may not be good for small precise areas to be glued. You can't spray a glob, then wipe/brush it into place. It won't wipe or brush smooth, at all. Best bet is to get adhesive from the pool table guys or a sports supply center. I've never used pool table adhesive, so ask for instructions on its use. Pool tables are large areas, edges can be taped off, so a spray glue can likely readily be used. Don't know if any instructions would accommodate small area applications.
If the only thing available is spray adhesive, then you'd have to tape off each drawer before spraying. There's a pool table supplier near me... 6-8 blocks away. I'll go ask them for some advise.
At any rate, when prepping for the felt application, do a dry fit (or 2) first and carefully, before committing to any gluing. You might want a dull knife-like tool to crease the felt into the corners and such. Use the creasing tool during the dry fit practice/procedures, also. Maybe do some practice gluings, with some scrap felt, on a small cardboard box.... see how things work, how to prevent wrinkles, etc. Thin felt may not always lay flat, as you work it, so practice to prevent wrinkles. Maybe get the (delicate hands) wife to help?
I don't think I do enough restoration to comment on refinishing that antique chest. I'll default to Bob.