Do mitochondria digest fats?
You are not far off the mark I would say. Strictly speaking it's not correct to say that mitochondria oxidize fat. By "fat" we usually mean triacylglycerides, and these must first be hydrolyzed into free fatty acids (by enzymes known as lipases) that can be oxidized in the mitochondria. But many researchers and textbook authors are not always careful with this distinction.

Most fatty acids are indeed oxidized ("digested" sounds strange to me) in the mitochondria into acetyl-CoA units, in a process known as beta-oxidation. The acetyl-CoA units produced are then further oxidized into CO$_2$ by the TCA cycle. These processes generate NADH and FADH$_2$ which is converted into ATP by the respiratory chain.

There are exceptions though. Very long-chain or branched fatty acids are instead oxidized in peroxisomes, which contain enzymes specialized for this purpose.

There is no such thing as "anaerobic respiration": in animals, respiration always uses oxygen as the terminal electron acceptor. Mitochondrial beta-oxidation is coupled to the respiratory chain, and therefore requires oxygen. Peroxisomal oxidation is not directly coupled to respiration, but the process still requires oxygen, which in this case is reduced directly by the fatty acid oxidases. So there is no truly anaerobic form of fatty acid oxidation.