Yes, cardio and other training can negatively impact your strength training. Whether or not this is the case really depends on the intensity of the other training, the quality of rest you get in between, the amount of calories you're eating, etc. To really determine whether this is happening, you may have to experiment with your schedule.
I prefer to distinguish between "cardio" and "conditioning". Cardio is typically just running, biking, or rowing some arbitrary amount of time or distance (long, slow distance) for the sake of daily exercise, usually with no specific cardiovascular goal in mind. Conditioning is carefully planned cardiovascular-intensive training intended to enhance or maintain cardiovascular fitness for some purpose.
Strength training can absolutely benefit from conditioning. But most people don't need to worry about conditioning until they graduate from novice to intermediate lifter (it sounds like you're a novice lifter, so this probably isn't a concern for you). Cardio (long, slow distance) is often overdone and often interferes with strength training recovery.
You should do as much cardio as you need to achieve your conditioning goals, but not so much that it negatively impacts your strength training recovery.
That said, it appears that your strength training is light on volume, especially with respect to legs and "core", which is probably why you're not seeing good results. Your strength training is also heavily weighted towards upper body work. A novice lifter who is trying to put on weight and gain general strength should really be squatting three days a week, deadlifting every other workout, benching or pressing every workout, etc. But you indicate that you only do "legs" once a week. If you're using a leg press machine or leg extension machine to do legs, then your back, abs, and hips are really being neglected. And if you're only doing legs once a week, you're not putting in enough volume to gain strength in your legs. In any martial art, your power comes from your foundation - your legs and hips - and is transferred via your core - your back and abs. The best strength training for your legs and hips also happens to be the best strength training for your back and abs: squats and deadlifts.
I think you would do well to start training compound barbell lifts three days a week instead of the split routine you're currently doing. Look for a novice barbell training program (there are plenty online). If you can't get access to a barbell and a squat rack, at least try and find a Smith machine to do squats in (the Smith machine is inferior to actual barbell squats, but it's better than a leg press machine).
If you're doing compound barbell lifts with enough volume, the next step is to ensure that you're eating enough protein to support gaining muscle mass, and that you're eating enough calories to support your plyometrics and Kendo training while also supporting your strength training recovery. You will need plenty of protein and plenty of carbs, and more total calories than you're currently eating if you want to gain weight.
To make maximal progress with strength training really requires complete rest. If you're doing other kinds of workouts in between lifting days, then you're not going to be making maximal progress. However, you can still make progress. I would only be concerned if you were making zero progress with your strength training, in which case you should examine your eating and sleeping habits, and potentially look at cutting back on your cardio activities. You might have to look at cutting back on plyometrics. I assume Kendo is an important hobby of yours, so I definitely wouldn't consider cutting that at all. The stress of Kendo training may make your strength training suboptimal, but that's no reason to abandon it, especially if Kendo is one of your motivators for gaining strength.
Also, I would recommend truly resting on the weekends.
