This is a multi-part question, so I will only be able to consider each of your questions in brief.
Lipolysis simply means the breakdown of fats in the body, and triglycerides are a kind of simple fat. Triglycerides are stored as oils in the seeds of many types of plants and in specialized "adipocytes" in animals. Lipases are enzymes that catalyze the hydrolysis of stored fats like triglycerides. They do this by hydrolyzing the C-O bonds that hold the three fatty acid chains to the glycerol molecule that is holding them. This hydrolysis releases the fatty acid chains for use in other parts of the body. Fatty acids are high-energy fuel molecules which provide energy for a variety of cellular processes.
Protease is a special kind of enzyme that breaks down proteins. It is usually found in the cytoplasm of the cell. Proteases break down proteins by catalyzing the hydrolytic cleavage of peptide bond, which hold individual amino acids of a protein together. Various proteases fragment only specific polypeptide fragments, as they utilize chemical reactions that have a specificity for only particular peptide bonds. For example, the protease trypsin cleaves peptide bonds only between Lys and Arg residues on the carbonyl side of the bond.
Urea is one of the end products of amino acid deamination. Amino acids which become deaminated degrade to yield pyruvate and citric acid cycle intermediates, with ammonia (NH₃) as a by-product. Ammonia is then transported to the liver, where adipocytes convert it to urea (CH₄N₂O) for excretion. The process that converts ammonia to urea is called the urea cycle. This cycle consists of four enzymatic steps. The breakdown of amino acids produces a number of compounds which undergo reactions to form the molecule citrulline, which passes into the cytosol of an adipocyte. Citrulline then produces another molecule called argininosuccinate via a citrullyl-AMP intermediate, which uses ATP to bind a molecule of Aspartate to Citrulline. Argininosuccinate is then hydrolyzed at the CH-NH bond, breaking it into two molecules of Arginine and Fumarate. Fumarate will be recycled into the citric acid cycle, whereas Arginine is further hydrolyzed, this time at C-NH bond, to form urea. Urea formation is important because it converts harmful ammonia molecules (which are attached to citrulline—the first compound in the urea cycle—at an earlier stage of amino acid deamination) into harmless molecules of urea. These molecules are then transported to the bladder, where they are excreted away.