Rounding dowel ends is easy; it is one of those things that is deceptively simple to do. You don't need any power tools. First, cut the dowels to length. You need to use a good sharp saw and cut them cleanly and square. This method will not fix a lot of tear-out. Using an 80 grit sanding block take the axle by the tip. Holding the other end against the sanding block rotate it dowel on a large circle allowing it to flex so the corners will get sanded off. In short order, you will have a rounded tip on your dowel. It takes a bit of practice, but you will soon get the hang of it.
For rounding edges, I highly recommend a sanding mop mounted in a drill press. There are lots of DIY videos about how to make them, but you will ruin a bandsaw blade, and those can be expansive. Good sanding mops are not cheap. I use the one from Klingspor Woodworking Store because they use high-quality sandpaper and you can buy replacement parts and refill kits. I think mine is 240 grit. This last a very long time. I have had mine for two years and have never had to replace the sandpaper. My sanding mop is one of my most used tools. Sanding mops work great for puzzle parts and getting the fuzzies off of the back of fretwork.
For holding toys while you sand them pad the jaws of your vise. I use leather stuck to the vise jaws with double-sided carpet tape. You will be amazed at how much easier it is to sand.
They make 1/8-inch router bits. It will work much better for toys than a 2/4-inch bit. To get smaller, you will need to go to 1/8-inch shank bits which will require a rotary tool configured as a router. There are some excellent quality carbide bits available with 1/8-inch shanks, but you will not find them at your local hardware store.
I used a router to round the edges on toys for a while. I didn't like it. It is easy to damage a toy with a router. Damage can be the point where it is easier to make another than to repair. It's also pretty scary routing small pieces. Routers make a mess throwing bits of wood all over. You also have to count the cleanup time. Unless I want the look of the routed corners, I don't route anymore.
To speed up hand-sanding use your vise. Use sanding sticks (aka fingernail files) you can buy scraps on line. Mke sanding tools to fit the task at hand. Dowels with sandpaper wrapped around them in various sizes and grits. Popsicle sticks (aka craft sticks) and with sandpaper on them work well. Sharpen the tip of the sanding stick to get into tight places. Use high-quality sandpaper. My favorite is sanding belt scraps. Don't skip grits, If it needs 80 grit use 80-grit and work your way up through the grits. You will wear yourself out trying to sand out defects or scratches with 240 grit sandpaper.
Sand your wood before you put the patterns on to cut the toys. Get a handheld orbital sander and use 80-grit sandpaper to smooth out the wood. Sanding first is much easier.