I started writing a reply this morning for you, and then my phone crashed.
I'll start with you BMI, which I have just calculated.
Your BMI based on your current measurements equates to 17.3 (this is calcuated by 168cm which is equivalent to 5'5, and 110pounds which equates to 49kg). Not only is this well below what is healthy, it falls into the category that would diagnose you with anorexia nervosa, if other symptoms aligned as well. Obviously your weight doesn't just define anorexia nervosa.
On BMIs for a causcausion female, the optimum BMI is between 20 and 25. Not 18.5-25. The lower band (18.5-20) is added to include the Asian population, who have a lower bone density. The difference between your current BMI and where you should be is quite significant. You should ideally be a weight of 56.5kg which equates to your original weight of 125 pounds. That would give you the very bottom of your BMI range at 20, which still gives you room to eat and gain whilst still being healthy.
Without sounding like a knowitall, you have quite clearly calculated your BMR incorrectly. A daily intake of 1350 is extremely rare, and that would require you to be completely sedantry like on bed rest for pretty much the whole day, so that you aren't burning calories. You wrote that you're 18, and at university. Therefore you will have a pretty active lifestyle, even if you aren't out partying or participating in much else all the time, you still require extra energy for studying and interacting with people. Therefore you will be needing much closer to 2000 calories.
So why then is your body increasing and decreasing so much?
Firstly 1200 or below is starvation mode. Why has every fad diet failed to stand the test of time (several decades?), because they are all below 1200 and none of them work. Any weight loss prescriptions for people who actually do need to lose weight are more than 1200 or on it, NOT less. This is because below, you send your body into starvation mode and do more harm than good. Your vital organs, including your metabolism goes effectively into "safe mode" and starts conserving energy because quite literally there is not enough. Out of the calories you require, 85% goes towards keeping your organs going, and your body functioning...forget any weight changes and activity...that's only 15% of your intake. So if you reduce your calorie intake to (pardon the pun) to eat into the 85%, your body has to get its fuel from somewhere.
Your metabolism slows down, so it won't work as efficiently as it did before. Therefore when you do eat it holds onto the food rather than processing it as efficiently as if it had done if you were eating regularly...therefore you will see a greater fluctuation on the scale.
If you eat regularly throughout the day, your metabolism works better, and you don't see inconsistent results. But...you have to be out of the starvation band...
There is another side to this as well, looking at what you've been eating and how much.
If you've been eating 110 calories, or even 700, you cannot be eating much in the way of carbs, which are essential for maintaining your weight, and preventing binges.
Binges aren't about being out of control, and undisciplined they are a sign that you aren't giving your body the right fuel on time. You body requires both oxygen and sugars to survive, so if you deprive your body of air, e.g. hold your breath, your will be out of breath while your body tries to catch up. The same is with energy. If you deprive your body of energy e.g. bread, oats, rice and so on (these starchy carbs are long term energy sources) your body will seek the most energy dense foods that it can find. This is how the body works.
My advice would be to increase your calorie intake to a band of 1300-1400 on a regular basis. Ensure you eat 3 x a day and maybe include snacks.
I'm currently working with my dietitian slowly to increase my weight, so hence I'm eating well under my limit. She's been increasing my calorie allowance bit by bit, to help me adjust. Currently I'm on a band of 1400-1500 (I expect when I see her on Friday it will increase again).
My meal plan tends to include:
Oats so simple + fruit compote (homemade) (around 230 calories)
Cereal bar as a snack/banana (100-120 calories)
Soup + bread roll (350 calories)
Snack (as above, the opposite of what I had before)
Dinner: Could be anything, but I'll try and aim to have some sort of carb, although this I find difficult. (Around 300 calories at present).
Night snack: hot chocolate with skimmed milk (200 calories)
I don't binge any more, although tend to need a snack if I find I haven't managed a roll with lunch...this completely validates my argument above.
My moods, and anxiety are much better, and my weight is completely regulated. Yes it is increasing, because I'm eating more than before. But the increase is only small. When I reach my target weight alongside an intake appropriate for my activity level (which will probably be above 2000 calories) my weight will fluctuate, because weight does that 1-3kg, even over a day depending on anything from water to sweat and so on, but it will stabilise.
The point about when you eat dinner, is a classic sign of body dysmorphia. You cannot have gained weight after eating a takeaway..unless you have eaten 21000 calories worth of takeaway = 1kg weight gain. Even that wouldn't actually show.
It sounds like even though you might not have an eating disorder, some of your feelings you need to discuss with a health professional. You certainly demonstrate a long term unhealthy relationship with food, and I would recommend speaking to your GP about this, who might be able to assist you with some dietitic support.
Hope this helps.
Kat