Would the polymers of proteins and fats cross the cell membrane? Would monomers of these polymers cross the membrane?
First let's talk about the chemical makeup of cell membranes. They are composed of long sheets of phospholipids which contain a polar phosphate group called the "head" and a nonpolar fatty acid chain called the "tail". These phospholipid sheets are arranged as bilayers, meaning that two of them are oriented against each other such that the nonpolar tail portions are facing inward toward each other and the head portions are each facing outward. This means that the interior of the cell membrane is a highly nonpolar, hydrophobic area, so nonpolar chemicals will tend to pass through a membrane easier than polar chemicals.
Now let's look at proteins and fats. Proteins are polymers that are composed of amino acid monomers. Proteins themselves have lots of hydrophobic groups and space, so they can pass through a cell membrane. In fact, most membranes have proteins embedded in them to facilitate transport of polar substances like ions through the membrane. But the individual amino acid monomers each have a polar carboxylic acid and amino groups, so they would be less likely to pass through the membrane.
Fats really aren't polymers at all. They are composed of a central glycerol molecule that has three long chain fatty acids attached to it. As a result, fats are extremely nonpolar and can pass through cell membranes. The individual fatty acid groups themselves can also pass through, but glycerol by itself is more polar so it is less likely to pass through the membrane.