Precomment 1: You mention under point 3, testing, that you are the assistant, not the course leader. So, to be practical, many "fixes" in the class will not be things you (as an individual, versus the archetypal you) can implement. We can discuss them in general (for other instructors). And you can think how to do better when you are in charge. But don't kill yourself over not being able to fix everything. That's just life and the World. Work on what is in your control--plenty to do there.
Precomment 2: Teaching engineering students is so much more prevalent than teaching math students that you should mentally think of it as the base case. Not "how to do it different from teaching math students". But how to do it. Then worry about how to change it for a math majors class! And if the class is mixed (likely), the math majors will be a tiny minority, and you should ignore special things for them. Don't worry, they will have analysis as a majors course, later. And they have other classes like chem or English to keep them occupied also. Not the end of the world if they have a standard calc course.)
First bullet, noise:
Don't make threats, you don't carry out. If you say you will lower grades, then just do it. Really, even a very tiny point deduction (1 point from semester of 500+ points) will change student behavior. Have seen it with lab google rules in chem lab. Don't even have to raise voice or debate. Just let them know the sword fell.
Another idea is just to eject a noisy student from class. Only one, very occasionally, will be dramatic.
Other than that, I recommend to discuss behavior at beginning of class (too late now for first lesson. But make it a point of a few minutes at start of a class, now.) The key thing to emphasize is that it is not fair to classmates and that it is wasting everyone's time. Don't open to debate or ask for suggestions. Just tell them, it's a problem and to stop it. You don't have to be angry or make threats. Just be direct and in control.
Also, I think having a loud voice is helpful. In the Navy it is called "command voice". It doesn't involve screaming, is not unpleasant in that matter--it is just using projection like an actor or a singer. (It's easy...just try it.) In addition, be animated, move around, command presence and keep it interesting!
Second bullet, low grades:
I don't completely understand your point, but I think you are saying the students are performing poorly on tests and you are using homework to help increase the scores. (I bet TA gets stuck grading those, haha.) In general, you need to try to make the tests more reasonable (point 3) and ALSO to improve the level of ability so the students do better on the exams.
Different points of view on grading homework (pro/con) but if you are going to do it, I would not do it for the purpose of raising grades in the way you are here. Instead do it to drive practice and drill (for the purpose of improving test performance). (Probably a low point value, more frequent turnins--every class for instance, and rapid returns.) Really, I think it becomes too much of a logistical hassle, especially in college.
I also find college classes have too few tests (major stakes tests only, which encourages goofing off and cramming). If you can do more tests during the semester, better. Yes, this loses some lecture time, but it is highly valuable practice--not like the lectures are perfect anyways.
Third bullet, exams too hard and mathy:
Cut that out. Use a reasonable text like Thomas Finney and then make the test questions close to the book. These engineers need to get experience taking derivatives and doing integrals and lots of algebra as well. That helps them with their homework in fluids class and with following derivations in engineering majors classes and textbooks. Spending time on baby real analysis is a waste if they don't have the basics yet. Same thing with super hard trick problems if they don't have the basics yet.
One little practical trick is to literally make the exam questions from the unassigned homework problems. (It encourages extra drill by the students.)
So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review.
I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.”
Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.
What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.
Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors.
Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!
[IOW, have some sympathy to building the muscles needed for getting them through chemE majors courses. Rates of reactions, bleed and feed mixing, fractional distillation, etc. You don't need to do examples using their course concepts...that makes it harder, since THEY don't know them yet and it is all word problems. But some sympathy and awareness of what is coming at them--and how calc class helps them--will help inform your teaching. You may even find some motivation leaking through to your students...]
Fourth bullet, poor HS prep:
You can't fix HS prep. You also can't fix native intelligence. You play with the cards given. But I actually think things are not as bad as you think. In general HS students are getting more math prep than several decades ago. College algebra and trig are common in HS as is some calculus exposure itself. Yes, there are some things that may have changed for the worse (TI time, atrophy of analytic geometry). But in general kids are getting more HS math exposure. Also, we send a lot of kids to college now. But if they are chemEs, they have the brains to get through a calc class if they buckle down and if it is taught in a practical (i.e. pedagically kind) manner.
One practical suggestion is SHOW THE ALGEBRA. I had this tiny crit of a comment for Mike Pierce where he did a great job showing with a u substitution, the endpoints change (in writing), but he glided over the change of the differential. I.e. du=2xdx if u=x². Really these kids can get a lot of practical exposure and practice in algebraic manipulation in calc class. I was a kid who had poor algebra background, but all the drill of calculus class ended up building up my algebra chops also. Didn't really need a remediation, just doing the homework problems (a gazillion of them) ended up building my algebra muscles as I did calculus. So show the algebra. When you are Carl Bender teaching advanced methods for seniors and grad students, you can glide over some things (but I actually still find him relatively kind). But if you are teaching calc 101? Show the manipulation!