Note: This came up in my "hot questions" feed - I didn't notice how old the question was until after I answered it. Sorry if this isn't required anymore or digs up a dead and buried debate!
Studies (and my own experiences) would suggest that using more words than symbols (to start with) would help everyone.
There's a page Here that deals specifically with guidelines on how to write mathematics (and consequentially, logic). There's an awful lot there, with examples and preferences over many different cases, and there is a list of sources to papers and studies at the bottom.
The general gist of the page is that mathematical language has a tendency to be too succinct when it comes to beginners - professional level communication (which assumes knowledge of the subject) is too often used with beginners. Words are left out, context isn't given and a glut of symbols is used where words could be more clear, if not more descriptive to someone with professional knowledge of the subject.
If you're only interested in literature to back up a point of view, stop here. If you want some more input and specific answers to your questions within your question, feel free to read on, but be aware that the rest is more opinion based on personal experience.
This backs up my own personal experiences on the subject - I studied Maths in university and had a module on Mathematics Education. Our lecturer said she has started the course because there weren't enough good maths educators in the world; either you understood the language - and then found it hard to understand when someone else didn't - or you didn't, and while you could then empathise and help someone else who didn't, you never truly felt comfortable with it. People who both "Got it" AND could see how to explain it to someone who didn't are incredibly rare, at least in the western world - the amount of people out there convinced they "can't do maths" is proof of that.
All this being said, it is still extremely important to learn the symbols properly if you are going to converse in the mathematical world. While it might be best to start with words to encourage understanding, at some point that understanding needs to be extended to the symbolic representations of the same concepts - learning the language of maths is in many ways just as important as learning the concepts themselves.
To address the questions in your question directly (and succinctly - if you want me to expand on any let me know, I'm just way of the length of this answer):
- Are there studies about how emphasizing one of these types affects the way students learn?
Yes, at least partially - see the link at the start of this answer (This one!)
- What harm does it do if only one type is present in teaching?
In most (those who don't "find maths easy"), only teaching the symbols will prevent true understanding and in many cases any understanding of what the student is learning. Only teaching the words prevents students from learning the language they need to converse in the mathematical community and will also ostracise those students who are gifted at maths but find language hard (quite common).
- How important is it that students can translate statements between these two languages?
I would say extremely important, but that's a personal opinion. For one, it shows sufficient comprehension. If the student can convert a symbolic statement into language, it shows they understand the concept correctly, and if they can convert from language to symbols, it shows they understand the symbols correctly.
- What are some key skills having to do with formal logical statements that I should make sure every student learns?
I don't feel fully qualified to answer this, as I'm not a trained mathematics educator (I've just studied the stuff), and I cannot claim any list I make to be remotely definitive and may even be vastly misleading due to glaring omissions. All I can say is that you're on the right track with this question in the first place - communication and clarity are key in logic and maths. If a student learns to phrase their statements in a way that makes the content as clear and obvious as possible, they've done the hard part.
- Is it easier or faster for students to read statements in words than in symbols?
for an absolute beginner, words will always be easier as they know what the words mean. As they do more, learn more of the symbols, the ratio of words:symbols that they find "fastest" and "easiest" to read will tend more towards symbols than words, because words take longer to read, but the actual ratio will likely depend on the person. For me, the below is easier to read than either of the examples in your question:
$∀y∈R, ∃δ>0$ such that $∀x∈Q : |y−x| < δ$
That's because I understand all the symbols and the connotations of the symbols that would take a lot of words to properly write out, but I still feel it benefits from a little English in the middle, just to give the "sentence" some direction. Of course, the advantage of pure mathematical/logical symbolism is that it transcends language; if I can write the whole thing without using any English, it will be absolutely clear to a French mathematician, which has its own benefits (and is another reason to learn the symbols!).
In conclusion, I would say that studies support the concept that words are (for the general population) far easier to understand (at least at first) than symbols, and that if your aim is to help spread the love of maths to more than the disappointingly small portion of the population that currently understand it, then advocating a "words, then symbols" approach would probably be vastly beneficial, even to those who would understand it without the wordy approach, because it may help them get their ideas across when communication with symbols breaks down.