"My family was going to get rid of this chair because of the mold problem on the back. First, how would I remove the mold and restore the condition on the back. Also does anyone know what vintage the chair is? 1960's? Is is even worth repairing and restoring?"
Hi Faith, the main thing to remember about removing mold from wooden furniture is to start slow and work up, and also to not saturate the wood. First, we definitely suggest you do this outside, and preferably on a bright sunny day with little humidity. Sunlight aids in oxidation and helps eliminate mold. You could try wiping down the wood with a soft cloth and a solution of water and mild dish detergent. Keep the cloth barely damp and rinse the cloth frequently, or use several cloths. You can also use a commercial wood cleaner, and/or mineral spirits. After the wood is cleaned, it is highly suggested you use a wood furniture wax to protect it. Here is a site with more tips.
As for the style, we agree with you assumption that it's probably from the 60's. Based off the photos, it doesn't seem like it would be too hard for an upholsterer to switch it out. Whether or not it's worth saving is really up to you. If there is a strong sentimental attatchment or if you really love the style then definitely hang on to it!
Does anyone have any furniture mold removing tips/experiences and/or suggestions for Faith?