Quite a few questions in you question so i'll break down what you told us and answer each based on your specific goals.
Your diet is fine for cutting, you aren't going to progress much on your lifting weights with 1500 calories and 1g/lb bodyweight protein. Very little carbs to provide you with energy though which will compound your issues with progressing on lifts in future.
You could be more optimal with your current program to maintain muscle. Something like Greyskull LP would probably be of use as it is balanced to most aesthetic goals and is 3x per week with reasonably low volume (3x5). Do whatever you want to do but: MAKE SURE YOU HAVE 1 REST DAY A WEEK MINIMUM.
Does cardio impair muscle gains?
Yes. No. Maybe. Despite the general attitude that "cardio kills gainz", this question - and thus its answer - is far more complex in nature. While resistance training and endurance work have many overlapping benefits, they also have many distinct and often times conflicting adaptations. (1)(2) Simply put, you cannot be an elite bodybuilder AND an elite marathoner. Again, there is a matter of degrees involved here, with the ultimate answer depending on your training status and the modality, volume, intensity, and timing of your endurance work (3).
If general health and fitness is your main goal, then including both forms of exercise is a good move. If maximizing muscle gain at the expense of all else is your primary concern, then endurance work should be minimized or eliminated entirely.
What cardio should I do?
Run, swim, cycle, play sports, whatever you ENJOY. If you enjoy it, you are more likely to put effort into it and will continue to do it. Don't force yourself to run if you prefer to cycle.
What are the different types of cardio?
- HIIT - This is short duration, very high intensity training that is distinguished by a high heart rate followed by a recovery and repeated several times. This has the effect of training your heart and lungs to deal with spikes in effort. In my opinion, this is what prevents heart attacks. It also has the side effect of influencing body composition so it's a good choice if someone wants to lose fat and gain muscle.
- Conventional/LISS - running cycling, etc. This means getting your heart rate to a given level and keeping it there for an extended time. This is good for stamina and getting better at whatever it is you're doing.
- Long slow easy activity - walking, sports, hiking, swimming, etc. This is stuff that is fun and relaxing. It burns some calories but mainly it's fun and relaxing. This improves your whole well being is subtle ways, and walking a comparable distance has been shown to be just as beneficial for cardiovascular health as either LISS or HIIT.
Should I balance my exercise with my calorie-intake.
Yes, sort of. If you are doing 3x 30 minute runs a week you probably don't need to adjust by much. You add that many calories back in through a very small amount of food.
I personally run 3x 10k runs a week and sometimes one will be longer. Based on my heart rate monitor, I burn approximately 1000 calories per run. As a result I need to eat some carbs beforehand to ensure I have energy to perform the run and get a good time. I know this because I did some runs without adapting and suffered because my diet is low-carb and I do intermittent fasting. The point of all of this is that you should not adapt your calorie intake to account for 'calories used' number from a machine. You should listen to your body and adapt to its needs. It's not as cut and dry as taking the number of calories burnt from the treadmill and eating that much food to balance it out. Only you will know what is the right amount of food to lose weight and have enough energy to do exercise.
Once you start running your weight will probably drop more than usual over a week or two, this is normal because your body is not used to it and will massively over-compensate.
You'll probably also get DOMS, you can workout with DOMS, just do a gentle warm-up and they will melt away.
Remember to hydrate during the day a lot so that you will not need to re-hydrate too much after exercise.