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Yes, you can. First clean the counter top thoroughly with a detergent. Then wipe the countertop with denatured alcohol to remove any residual oils or grease. The next step is to sand the entire surface with 120 grit sandpaper to rough it up slightly. Don't gouge the countertop... just scratch the surface a little to help the paint to stick.
If there are any serious chips in the surface, they should be filled with a two-part wood filler called Minwax High Performance Wood Filler. This product sets quickly, sticks like crazy, is very sandable. Though these characteristics are important, the most critical quality is that this product dries as hard as a rock… important for a surface that has frequent contact with hard objects. Once you mix and apply this product, do your initial sanding no longer than fifteen minutes after it becomes solid. If you wait too long, it will become so hard that it will be difficult to sand by hand. By the way, this product is very similar to auto body filler, which shares its strength and handling characteristics.
It is important to apply a primer/sealer before your finish coat. The primer/sealer will stick better to the laminate than the finish paint, decreasing the potential for paint chipping. I would suggest using an alkyd (oil) based paint, applied with a short nap roller for the smoothest surface. Also, tint the primer to match the finish paint if the finish color is anything but slightly off-white.
With oil paints, two thin coats are better than one heavy coat. The other advantage of an oil paint is that you can sand chips and nicks smooth when it is time to repaint (sorry to interject that little reality). With a latex paint, you will get a less smooth surface initially and chips cannot be sanded smooth.
One caveat… don't expect this paint job to equal the strength and durability of the original laminate. It won't even be close! You may have to change some of your bad habits (if you have any). For example, always have a cutting board available, place very hot items on protective pads, and keep the cats on the floor (OK, I know that's impossible, but I am a hopeless dreamer).
There are stronger paints available than alkyd oils. These are the two-part paints that are used by professional bathroom tub refinishers. Not as widely available as alkyd paints, they are relatively very expensive and designed to be sprayed, though they can be brushed and rolled. Just expect to apply two or possibly three coats. They are more durable for sure, and do not need a primer. Remember, their primary advantage is in their ability to stand up to the difficult conditions in a hot, steamy bathtub enclosure… as pleasant as that can be at the end of a loooong day!! You will have to decide if this alone justifies the additional materials cost.
Formica can be coated with many paints or clear finishes as long as it is absolutely clean and is very slightly sanded... 220 grit paper would be the roughest that should be used.
Since you want to maintain the appearance but just lower the gloss, you could try a water-based polyacrylic coating. This is a durable, abrasion resistant product that should serve your purposes. Minwax is one manufacturer of this product. Apply it with a foam brush or lint-free roller... very lightly! Multiple coats are preferable to a single heavy coat. This product can also be sprayed if you have access to the proper equipment. For a small area, though, spraying is unnecessary to obtain a smooth finish.
Thorough stirring is essential. These products are naturally glossy... additives are used to produce the satin or "matte" effect. Don't stir too vigorously and don't shake the can... the bubbles caused by shaking may not "pop" fully before the finish dries leaving you with a rough finish. Sand lightly between coats with 220 grit sandpaper. Do not use steelwool.
This is not a manufacturer-approved use for this product, so don't expect to find it listed on the can. However, the water-based polyacrylic products are recommended for use over old finishes including polyurethane… provided that the surface has not been waxed or treated with an oily finish or soap. Hence, you should get good adhesion on a plastic laminate.
Please don't get upset, but the bad news is that you must strip the polyurethane from the countertop and start over. Applying paint remover is the easiest method, and it does not affect the laminate. You should be careful near the edges since it will affect the contact adhesive. Also follow all the warnings regarding ventilation and physical safety on the label of the product.
You cannot apply polyurethane over any finish except virgin polyurethane. That is, polyurethane that has not been treated, sealed or waxed. Other surfaces don't give the polyurethane enough grip for it to survive for very long. If you use quality paint you do not need to apply any protective finish over it.
Latex paint is not a great choice for any surface that suffers possible abrasion, such as a countertop. I also loathe using latex paint on bookshelves or other surfaces that are going to have objects placed on them. Instead, a better choice is oil based "alkyd" paint. Alkyd paints are the finest of the oil paints. They dry very hard, are washable and about as abrasion resistant as you can get in a paint. The only better paint you could apply in my opinion would be epoxy paint, which is a two-part paint that chemically sets instead of drying. However, they are not as easy to find and quite a bit more expensive. I have used polyurethane paints a few times, but I am not convinced they are really any better than alkyd paints in durability.
With care and gentle treatment, the painted surface should last years. The surface will never be as tough as the original. Should you get some dings or scratches, the thing that most differentiates latex paint from oil is that latex paints do not sand well, so it is virtually impossible to smooth out those scratches or marks. Oil paints, on the other hand, sand beautifully and you can restore a smooth surface (prior to repainting) without any filling or paint stripping... again