Absolutely! You can and should shower and bathe as you usually do when you have your period. Most women find that during their periods, the best way to prevent unwanted odor (caused by bacteria rather than the menstrual blood itself), is to practice good hygiene. Washing with a mild soap and warm water will keep you feeling clean and comfortable.
There is nothing wrong or harmful with taking a shower while you are menstruating -- if there is any time you want to keep yourself a bit extra clean, that is the time. I used to bleed so heavily I’d put a tampon in before I left the shower stall so I wouldn’t drip across the floor or get blood on the rug or towel. But there isn’t anything dangerous or gross about it -- it’s completely natural.
Absolutely! You may even want to shower or bathe more often while on your period. I tend to take more baths just to relax and ease cramps. You might notice a little bit of blood dripping out if you take out your tampon or remove your pad before showering, but that's okay! It will just go down the drain with the rest of the water. If you want to keep your tampon in while showering or bathing, that's fine, too. If I keep my tampon in, I usually change it afterwards since it tends to absorb some of the water.