To quote Dan John:
If you don't know where you're going, any road will get you there.
You say you're training to get / stay healthy, which is great, but it depends on your definition of health.
Health is a state of physical, mental and social well-being in which disease and infirmity are absent (from Wikipedia)
The state of being well, without disease or dysfunction (from our own tag definition)
In a way, this leads back to your goals.
For me, looking at it in a purely physical sense (the mental and social well-being aspects are a different point), I would worry about my health if I lost the ability to perform certain movements.
Dan John (again) mentions that the inability to stand on one leg for 30 seconds is a sign of poor health, and has a few anecdotes to go along with it. Likewise, if you look at the older population, getting up from the floor is a movement that people seem to lose in ailing health (someone told me last night that the difference between older people and younger is that younger people fall over, and older people have falls, made me smile).
For myself, along with standing on one leg and getting up from the floor easily, I would add the ability to deadlift my own bodyweight for reps, do at least one pull up (this could be regressed to hanging from a pull up bar for 30 seconds if my shoulder injury flares up) and overhead press half my bodyweight for reps and do some press ups.
Why these? Because they're easy standards that I've held myself to for much of my adult life, and if I lose one of them, then it's normally down to injury or illness.
Is your current routine going to lead to any physical dysfunction or infirmity? Honestly, this depends on you. I couldn't do your "leg day" as I'm currently dealing with both an ankle and knee injury, the thought of jump lunges sends a cold chill down my spine. Likewise I wouldn't recommend your routine to someone who's heavily overweight, again, jump squats and jump lunges wouldn't be a great idea for an obese person who's never exercised before.
For yourself, assuming you're not injured or seriously overweight, and are used to the movement, then I don't see an issue with your current routine, even without any sort of warm up.
Should you stretch? Again, it depends on you and your goals. Typical desk jockey computer folk, like myself, tend to suffer from tight chest and hip flexor muscles due to sitting all day, so I stretch those parts daily. Once I've fixed my knee, I'd also like to achieve side splits, so I'll start stretching to work towards that.
If you don't have a flexibility based goal, and you have an active job, then you might be fine without any stretching (lack of joint mobility is another issue with the older population, so you should probably have a few mobility related markers for health, I occasionally check to make sure I can still touch the floor without bending my legs).
Is your routine good? I would say no, but I've seen worse.
I like to run through a variation of the primal movement patterns in my training, so I tend to go with or use variations of:
Hinge - deadlift or kettlebell swing
Squat - goblet squat in the warm up, I'm no longer interested in a heavy back squat, got a few too many injuries for that
Push - overhead work mainly as I do sit at a desk all day
Pull - pull ups and front lever progressions, because I'm also a climber
Everything else - Turkish get-ups, loaded carries, 1 arm kettlebell swings, bouldering and climbing
You address some of these, but I don't see any hinge or pull based movements in your routine (though you could argue lat dips are a pull variation).
I would say, pick some goals and work towards them. It might be working towards harder press up and squat variations (one arm press up and pistol squat, Pavel's Naked Warrior program), or a farther distance on the eliptical trainer while maintaining a constant heart rate. Whatever you choose, having something to work towards gives your training sessions more direction, and can make answering questions on here much more helpful and concise (though not if I'm the one who answers them ;)