I got an email that pretty much said "we want to hire you, but what is
your salary request?".
And I'm afraid that at this point I ****ed up. I asked for 20% more
than my current job, as I feel I am worth that, since I would be
working as a fullstack developer, but a week has passed without a word
from them, so I'm not having too high hopes about it anymore.
The fact that you haven't heard back for a week doesn't mean you f***ed up. It might mean that they will get back to you soon. Or it might mean that the company is attractive but doesn't pay well enough to meet your your requirements. That's not a failure on your part. Would you really want to work for them if they were only offering 20% less than your current pay?
You asked for what you think you are worth - in my opinion that is usually the correct approach. Now try to be a bit more patient and see where this goes. You may get what you asked for, you may be offered less than you asked for, or you may be able to negotiate more.
So if I'm not getting the job, I'm probably going to end up in a
similar situation in the near future, but how should I submit my
salary request?
Some choose to leave the future salary requirement open, and just refer to their current salary "I'm currently making $X, and feel that I'm worth more. I am open to hearing more about the benefits package and working with you to come up with an appropriate salary." Some don't even mention their current salary, thinking that it "anchors" the offer to a level that is too low.
I made it clear that it is very much negotiable, but should I just
make it clear from the beginning? I would've given an estimate in the
first application, but I didn't know enough about the company to make
an estimate.
Indicating that it is negotiable is often a good thing to do.
A company doesn't want to offer too little or too much. But a company also doesn't want to waste time with someone who is clearly outside of their budget.
I don't think you did anything wrong if you truly feel you are worth 20% more and don't want to settle for much less.
On the other hand, perhaps that most important factor isn't the salary, and instead the possibility of working less than 60 hours per week is your most important requirement. In that case, you may indeed want to settle for less. And in that case, asking for 20% more may have been a mistake.
Instead, you may have been better advised to emphasize that the hours (or working environment, etc) are more important, then seeing what offer you got.
For future negotiations, go in with a clear understanding of the importance of all the work variables - both financial and non-financial. Then you'll be able to weight them effectively, and get what make you happiest.