Along the same lines, I have a corner cupboard made by my g-g-grandfather around 1850, may be a little earlier, may be a little later. It doesn't need cleaning, but I worry about it drying out. It is mostly solid, but the raised panels on the lower doors have veneer in the center and a small piece has dropped/broken off. Don't want any more spots to do that. It may be the result of being hit by something-it happened years ago at my mother's house. Forgive the hijack, please Camlan. It seemed to go along with your question and maybe will give you more info, too, if anyone has ideas.

The coil cleaning detergents are very effective. They have different types (for indoor coil vs outdoor coil) Use caution as they are acidic (especially the outdoor solution). Warm soapy water, followed up with a warm water rinse down is effective also. Compressed air is a great follow up after utilizing the coil solution or soapy water first. Here is a link that might be useful: A/C Cleaning

The best way to clean furniture for painting depends on the current finish and condition as well as the desired finish. If you only want a distressed paint finish you may not need to be as thorough as if you want a very fine smooth painted finish. If the piece is really dirty I have used Murphy's Oil Soap or TSP, rinsed with the hose (on the patio!), and dried quickly with old terry towels. Then let the piece dry for another day, sand if necessary, and stain or paint. When DH painted our kitchen cabinets he used mineral spirits which removes grease. For an alligatored finish you can use denatured alcohol or laquer thinner to remove shellac or lacquer. Sometimes with old pieces there are years of wax imbedded with dust and dirt which needs to be removed or the paint won't adhere and mineral spirits or denatured alcohol will remove that. It's necessary to use good gloves and work in a well ventilated area as the stuff is quite toxic. There's a product called "Citrus Natural Solvent" I'd like to try as it's supposed to be more environmentally friendly than other products.

I've never refinished wicker but I have a couple of thoughts: The chair will be dusty - you are going to have to clean it throughly (maybe compressed air) , and wipe with a damp cloth. If the chair has been varnished - you can't restain it - but you can use a colored varnish. If the chair hasn't been varnished then you could use a tinted rubbing oil.