Your question has several parts to it so I will attempt to address them one-by-one.
In terms of protein stability and turnover, this is a nice paper describing how human cell (HeLa) line, although this is a cancerous cell line, is used to measure the turnover of proteins (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3316722/?report=classic). They have used labelled amino acid technique and mass spectrometry (SILAC) to make measurements of protein turnover. It is concluded that on average the turnover is 20 hours, which is correct in my experience when doing gene knockdowns in Drosophila (S2) cells using RNAis as we incubate the RNAi with cells for 72 hours to ensure most proteins that we targeting are degraded and we can see the desired effects in cells.
Now getting to your questions regarding organelle turnover, you have to consider that cellular organells are very precious to the cells as most of them consume a lot of energy to build hence they are maintained unless they suffer critical damage, and they would become detrimental to the cellular integrity. In that case organells such as mitochondria undergo a specific type of degradation called mitophagy, which is a subtype of autophagy. Now I'm sure there are specific average times for the so called turn-over of cellular organelle under "normal" conditions but thats a slightly misleading view since the turnover can be tipped through reactive oxygen species (ROS) or ageing leading to less efficient cellular maintenance and buildup of toxic aggregated proteins or ROS.
Now just to clarify the use of the word half-life in you last paragraph, half-life normally refers to the time it takes for half the molecules in a given system to degrade (or to be more correct Half-life is the amount of time required for a quantity to fall to half its value as measured at the beginning of the time period) so I do not agree with your use of the word half-life in the last paragraph of above question although human DNA does have a replication limit (Hayflick limit) of roughly 40-60 due to telomer shortening and afterwards the cells (due to the DNA) that no longer replicates will enter senescence and eventually die.
Now getting to non-replicating, highly differentiated cells such as neurons, it still remains a mystery as to how neuronal DNA remains so highly stable (although DNA is a highly stable molecule (compared to RNA) and its further stabilised through histones), while undergoing many dynamic processes such as transcription (since if anything goes wrong the cells cannot be replaced at least in CNS) and how axons of some neurone that can reach over a meter in length are maintained so well for decades. There is a lively argument amongst the scientific community about this, which you can find out about if you simply search for axonal maintenance in a search engine.
Hope this answers some of your questions.