The tool rest does not need to be adjustable for height. The height should be level with the centres. Some lathes are fitted with a simple bar rest that is attached to the stocks for support. However I prefer a scheme that allows more flexibility with regards to positioning. Described below is an independent tool rest that allows close positioning for effective turning.
A handle could be added to the wooden disk although I find I can tighten the tool rest sufficiently just by turning the disk. Glue leather to the bottom of the tool rest to improve the grip onto the bed. The tool rest is best made of a strong dense wood. You could use marine ply about 1" thick. The main thing is that it should transmit the force applied to the gouge by the work to the bed without distorting. Make the slot just wider than 12mm so that the rest can be adjusted easily. The brass strip is a refinement that I found useful. Recessing it into the top of the rest and epoxy resin have secured mine for two years.