OK. Let's go point by point.
1. If you've only had her one week, she has not really bonded with you, she is in what we call the 'honeymoon' period and, if she is that aggressive during this period, chances are, she will be even more aggressive once she starts feeling comfortable in her new home. Let me explain. The honeymoon period is, essentially, a survival mechanism. Basically, the bird is in its best behavior because it's completely unsure of the new humans and the new home so they behave the best they ever will in order not to create friction or call undue attention to themselves. Once they start feeling more comfortable, they become much more assertive -and that translates into nips and bites if the bird decides anybody is not doing what they should be doing. This does not mean that she will not bond with you. It just means that the behavior you see now is not what you will get later on.
2. Yes, she is overly hormonal. The rubbing you mention is masturbation and birds only do it when they are 100% sexually frustrated so you need to get her endocrine system back on track and that can only be done through a strict solar schedule which does not mean covering her cage when it's dark outside but exposing her to both dawn and dusk for, at least, one hour. This means that her cage needs to be near a window (so she can get the twilight) and that the lights overhead should not be turned on until the sun is completely out and shining into the room through the windows. In the evening, it's the opposite, you turn off the overhead lights when the sun is halfway down to the horizon and allow night to fall naturally - then, once it's completely dark outside and the bird is asleep in her roosting perch, you cover the cage with a black-out material (this is only necessary if the cage will be exposed to any kind of light, whether it's from another room or from a street lamp coming in through the window). This time of the year, the senegals are hormonal because, in their natural habitat, they breed in the fall, when the days are shorter and the food is plentiful after the summer rains but, in captivity, if we don't regulate their diet, they become hormonal also this time of the year.
3. Light. Yes, you need a special light but NOT a 'bird' light because all the lights labeled 'avian' are nothing but reptile lights that have been relabeled and which have a too high Kelvin Temperature (this makes the light too red and it brings them into breeding condition). You need to get a full spectrum light that has a KTemp between 5000 and 5500 (but I would get a lower one for now so you can bring her hormonal production down a bit) and a CRI as close to 100 as you can. I am now using Dr. Mercola's but, as I change them often (you need to change the bulbs every 6 months because they 'lose' the full spectrum after this time), I do research every time and, if I find a better one, I get that one (I have switched lights many times because manufacturers would, sometimes, produce one that is great for us and then stop).
4. Diet. The food you are feeding is no good. You can't free-feed seeds. Period. I am sorry but there is no two ways about this. If you do, the bird will end up with liver disease and nutritional deficiencies. Mind you, I am not saying that you should not feed seeds (I do it myself) but you can't just fill up a bowl with seeds and just leave it there. Parrots diets in the wild are not just seeds. They do eat them but these are 'green' seeds (the seeds inside the fruits) and they have to eat the whole fruit first before they get to the seeds. You can choose from a number of foods: chop, mash or gloop. People now use whatever name they want but they used to mean different things. Chop was just fresh veggies and fruits chopped up and frozen in individual portions (people use ice cube trays). Mash was the same thing but put through a finer chopping in the food processor (I never even liked the idea of a veggie/fruit puree for parrots). I use gloop which is cooked whole grains and pulses mixed with chopped veggies or whole veggies that are small in size (like peas and corn, for example). So, in the morning, about one hour after dawn begins to break you feed your bird the healthy food (I serve gloop with one fruit, one veggie and one leafy green but I make it a different one every day) and, in the evening and right after you turn off the overhead lights, you take away this food and put a measured portion of a seed/nut mix (and when I say nut, I don't mean peanuts which are not nuts at all but actual tree nuts like walnuts, almonds, macadamia, etc). For a Senegal, it's about 1/8 of a measuring cup (mine get a couple of nuts -like one almond and half a walnut- and the rest in seeds but the seeds come from a cockatiel mix that has a bit of safflower, a few striped sunflowers and mostly grass seeds). If you don't know what she ate in her previous home, you should find out because, if it was seeds and little else, she is going to need to get a multivitamin/mineral supplement for a couple of weeks, at least, to replenish the lacks. And you should also concentrate on feeding her food rich in betacarotene (they need a lot of vitamin A but you can't give them the final form -too dangerous for their liver- unless they are deficient in it, you need to supplement it by feeding the precursor: betacarotene).
5. Training. Yes, of course you can train your Senegal but I would wait until the honeymoon period is over and the bird is on its way to becoming bonded with you. Why? Because it will work out better for you in the long run. Parrots can only be trained through rewards - whether this reward is a high value item (this is a food item that the bird loves and which it does not get at any other time but when it obeys the command) or through praise. But, in order for the praise to work, the bird has to love and want to please you. They are different than, say, dogs, for example, because parrots are not hard-wired to be eager to please (they don't belong to hierarchical societies where the lower echelons want to please the alphas) so you need to establish a strong bond of love for them to want to do anything for you. As to train Karmi not to react with bites right now - well, the thing is that you can't train a bird in pain not to have pain and you can't train a bird that is sexually frustrated to become 'unhorny' . Birds are not like mammals, their sexual organs are active half the year and inactive the other half. When the daylight hours reach a certain number (which has been pre-determined by the conditions of the natural habitat -meaning when there is plentiful and rich food and good weather), they start producing sexual hormones, their gonads become active and they start to grow, preparing for reproduction. BUT when you keep a bird with long days and too rich food all the time, the poor bird's body doesn't know when it's the time to stop producing the sexual hormones and continues month after month, year after year. This creates all kinds of problems because the gonads grow and grow and become so large that they start displacing other internal organs (which starts as discomfort but becomes pain that never stops) and the poor pet bird, which has no mate, has no way of relieving the intense sexual desire (and that's when they masturbate). Birds that are kept at a solar schedule and with a good diet go through cycles -they start producing sexual hormones but they stop after a few months- so there is never any pain or frustration.
6. Flight versus clipping. There are many problems with depriving a bird of its only natural mode of transportation as well as the one and only predator-avoidance mechanism... from a feeling of chronic insecurity (it can't get away from danger) to physical problems (their respiratory system health is directly linked to flight so, when they don't, there is a pair of air sacs that doesn't inflate and deflate properly becoming, in time, atrophied which, in turn, makes fertile ground for infection) but the other one is that nature did not give birds any other form of exercise. It's like a person in a wheel chair and, if you know one, you also know that it's not only a matter of them not walking, it's decreased intestinal motilily, it's urinary tract infections, it's not sleeping well at night, etc. For birds, it also means that there is no fast way for their body to get rid of bad hormones (like stress and sexual hormones). Of course, there is nothing you can do now and, if you decide to allow her remiges (flight primaries, the long feathers in the wings) to grow back, you will have to wait for her molt but I am just telling you this so you also take it into consideration when it comes to her 'moods'. She is between a rock and a hard place, the poor thing!
And I bet after reading all of this, you are thinking to yourself: "SHEESH! And I thought a bird was an easy pet to keep!" Nope, they are not. I would rather have 20 dogs and 20 cats than 2 birds
Compared to dogs, cats, guinea pigs, etc. they are super complicated to keep healthy and happy and because most people have no idea, the poor things end up been rehomed all the time because they start biting, screaming, plucking, etc from the inadequate care... But, as difficult as it sounds, it really does become second nature after a while and you start doing it mechanically so it's not as if it's not doable, it's only that it means a HUGE change in the way we do things.