There is definitely a correlation between strength/fitness and your overall diet. An inadequate protein intake will lead to less than optimal recovery and progress, your body needs the amino acids to repair damaged muscle tissue from exercise and for hypertrophy to occur. An inadequate caloric intake will limit the amount of lean body mass you can build as well as impair your ability to recover from exercise. The reason why most people tend to lose a bit of strength on a "cut" is due to a few variables, but most of this strength loss can be minimized if you lose weight slowly and with a more moderate caloric deficit instead of the far more aggressive fat loss phases that people tend to put themselves through.
If your goal is to get stronger you will need to build up some muscle mass, there really is no way around this. However, noobs to lifting can make tremendous strength progess due to increased neural efficiency, meaning that their bodies are learning how to execute the movement and utilize as many muscle fibers as efficiently as possible. This can be compared to any physical movement that you are unfamiliar with, remember how wobbly and awkward you felt when you first learned to ride a bike. As for what rep range to use and how to train, you are better off sticking with a more moderate rep range (5-8 or so) as this is heavy enough to put your muscles under mechanical stress but will also provide them enough volume and stimulus to grow, if you do enough work sets. Powerlifters often train with very low reps with weights that are close to their max, 90%+, this will train their nervous system for a 1 rep max but will require proper programming, as this easily leads to nervous system fatigue. However, they will also incorporate a moderate to high volume of accessory work that will provide the stimulus to grow their muscle mass.
So, I think you'd be better off focusing on a training routine that is focused on a few compound lifts done in a more moderate rep range with multiple sets. Stop one or two reps shy of failure and focus on slowly increasing the weight over time. Consider purchasing a set of micro weight plates, as this will allow you to make a jump of as low as half a pound per workout. Full body lifting workouts 3 days per week and you'll get strong pretty quickly if you eat properly.
