I have 2 hedera helix in my office, different ages. The new one I brought in about a month ago looks fine, but the older one has developed sticky leaves, dotted sparsely with tiny tan salt-like things (they don't move) on the leaves and parts of the stems. Some of the leaves have shiny spots where it looks like the little spots have liquified -- or something. It's also dropping leaves. Any clues?

Your plant has spider mite. It's very common with ivy plants kept inside where the air is dry. I would try to drown as many of them as you can either by washing the plant off in the sink or by submerging it in a bucket of water. I've had sucess using neem oil mixed with water and a little dishsoap. I use the dynagrow neem oil which I order online, the instructions are right on the bottle. Neem oil is non-toxic, and is often used in beauty products in India.