Rather than just giving a short answer to each question, I thought I'd discuss the topic in a more general way so you can (hopefully!) see how all your questions are related.

To start, let's first look at the topic of buoyancy. According to Archimedes' Principle, the buoyant force on a submerged (or partially submerged) object is equal to the weight of the fluid displaced by the object. Here, we use the term "fluid" because this same principle applies to both liquids and gases. In essence, the buoyancy of an object in a fluid depends on the relative densities (mass per volume) of the object and the fluid. If the object is more dense than the fluid, it will "sink."
Conversely, if the object is less dense than the fluid it will "rise" or "float". For objects less dense than the fluid, the relative densities of the object and fluid determine what fraction of the object is submerged.

Question #1 and #4

To visualize this, let's imagine a wood cork. A wood cork in air will fall (or "sink") to the ground because a cork is more dense than air. In other words, the mass of the cork is greater than the mass of air that would occupy the same volume as that cork. However, if we instead place the cork in water, it will now "float" because the cork is less dense than the water. Hence we can see that the buoyant force exerted on the object depends on the density of the fluid. Let's now imagine that the cork was made out of lead instead of wood. In this case, the cork will clearly "sink" in the water since lead is much more dense than water. Hence, the object's material, or more specifically its density, affects its ability to float.

Question #3

The shape of an object does affect its ability to float. Let's talk about boats. Some, such as cargo boats, are made of steel. Clearly, steel is more dense than water, yet these boats still float. The key is that only the hull is made out of steel, while the inside is mostly empty. The shape of the hull allows the boat to displace a volume of water equal to the weight of the boat. Since much of the submerged area is air, the average density (total mass of the boat divided by the volume of water displaced) is less than that of water, thus allowing it to float.

Question #2 and #5

An object will float whenever the weight of the object is equal or less than the buoyant force. If the density of the object is less than the density of the fluid it will always float. An object made up of materials more dense than the fluid can still float as long as the average density is less than that of the fluid (as described in the answer to Question #3).