Great questions, Annette!
I'll start with the barlow. Most people think that a barlow is somehow a neutral or non-active optical element. Thing is there is no optical element that is neutral. ALL are active. This includes the human eye which has its own aberrations unique to every individual, and its own set of corrective capacity. Barlows can both exacerbate and mitigate aberrations, depending on the EP/scope combination.
As for which determining which EP design is best suited to which scope, there's two parts to this. One is knowing that most EPs are designed for a convex focal plane. But remember, this is not a hard and fast rule as EP designs are not geometric in their aberrations. Take the Plossl and Orthoscopic designs. Both are old designs (more than 100 years old). Their lens design/arrangement is the same in all focal lengths, including the glass type. What changes to give the different focal lengths is the thickness of the lens elements and the radii of the various surfaces. And this in turn means that there will be differences in performance between the different focal lengths across all the different scope types AND between focal ratios. Hang on there, I'm coming to a point... The above is to say things are not so straight forward...
Now, the plossl design was made for Newts. But when it was created, Newts were f/7 or slower. You will find that plossl performance is challenged as the f/ratio gets faster.
As for which other designs best suits Newts, this brings me to the second part of the reply. EP manufacturers will never say this. The reason being is as most people think a scope is a scope is a scope, they also think and eyepiece is an eyepiece is an eyepiece. So if a manufacturer were to say that their $200 eyepiece is best for refractors, people will think that there must be something wrong with that line and will not buy it. This leaves us amateurs kicking the can, spending our hard earned money, getting ****ty results with these eyepieces, and dismissing outright entire lines solely because of our ignorance. But all is not lost. It is with threads like this that the how and why of EPs and optics that shed light on the true relationship between EPs and scopes, and everyone learns how to recognize the tell-tale aberrations of optical mismatching. Astigmatism being the key one.
Remember, even between high end EPs, performance will vary between individual EPs, and will again vary between different scope designs and focal ratios, this includes with Newt specific eyepieces. A 20" f/4 Newt will have a focal plane of a different radius from an 8" f/4 Newt, and so EPs will perform a little different again. ALSO,
an EP line today is defined by all the EPs first having the same Apparent Field of view, and the same or similar amount of eye relief. The ONLY difference being focal length. So the internal design and composition of the different glass elements, and number of elements can differ tremendously. It is nothing like old EP designs, like the plossl, which was a four element design in two groups. Look closely at the spin on EP adverts today and you will read stuff like "EP design with 5 to 7 elements"... This is not a "plossl" design now, is it. Things are never straight forward.
Oh, and so you know, to recognize astigmatism, the stars along the edge of the field of view appear as little "seagulls", cocentric to the centre of the field of view. The more grotesque or larger the seagull effect, the more the optical mismatch. Coma makes stars appear as little comets radiating out from the centre. If all you see is coma, THIS IS GOOD! Seagulls not so good, but not necessarily fatal either. It also depends on your individual eyes and your preferences as to how much it bothers you. Recognize astigmatism being described in a review, and you will be streets ahead if the author of the review doesn't!
There is one more line you may like to keep an eye out for your Newts, the Vixen LVW line. Though now discontinued, they are now pretty much just second hand items. One brilliant thing about this line is they are one of the very few lines that perform really, really well across all scope designs! Yes there will be some slight variations, but as a whole they are very bloody good.
Oh, many people also think that the Baader Hyperions are a copy or clone of the LVW's. They are not. They are only similar in size and colour coding, but the Hyperions are designed for a convex focal plane. Their internal design is not any form of copy of the LVW.
I hope this answers some of your questions.