Where did you get 1600 cals from? Did you put your details into the NHS BMI calculator? nhs.uk/live-well/healthy-we... It's important not to eat too little.
No, low carb, high fat, moderate protein, eat when hungry, stop when full, fresh produce only, lots of veg, small amount of fruit, ditch the processed stuff
Remember rest is required for exercise to work, so alternate days would be better than every day.
As moreless said, you need to be replacing your carbs with fat. A low-carb, low-fat, high-protein diet is extremely dangerous: google "rabbit starvation". Humans are not carnivores: we cannot use protein for energy efficiently. You might initially see results, but you will quickly "crash" if you keep this up - you will most likely start to feel lethargic and get irresistible cravings for carbs and/or fat.
It's really quite depressing how the nutritionists are still banging on about fat->cholesterol->heart disease. This purported link has been repeatedly debunked, but the advice still hasn't moved on. If you join the LCHF forum there's a lot of chatter about this: cheritorrox just posted her latest cholesterol test results, and there is increasing evidence that older people with low cholesterol are at greater risk of stroke.
Oh .. no need to control your calories on low-carb diets. They work by fixing your appetite (or more accurately, low-fat high-carb diets fail by severely disrupting your appetite - there are a wide range of "healthy" diets that work fine, but low-fat high-carb is not one of them).
Apart from the conversations about cholesterol above, please look at this whole way of eating as a change of lifestyle. If you can't change everything at once then start by cutting out junk, then try to cut down other carbs (bread, pasta rice, spuds) and get into the habit of more healthy fat (e.g. ditch anything that says it's "low fat" as it's yet another food industry con!)
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