I'd say you should just start learning it and see for yourself whether it works for you or not. It's not like once you decide the rest of your life is set in concrete and your fate decided for all eternity. It is possible to try it out for a month and at the end of that time assess for yourself whether you're satisfied with the results... Then you can base your decision on your own experience rather than rely on the advice of some stranger on an internet forum who could be much smarter or dumber than you, or maybe even a deceitful troll...
But, I'll make a few comments anyway:
•"I'll keep learning English until I become a fluent speaker. After accomplishing all of my English-related goals I'll pick another language."
You seem to write English quite well already. What are you waiting for? If you're not satisfied with your English yet, you're probably one of those eternal perfectionists, and thus never will be satisfied. You'll probably never consider yourself a fluent speaker and will always be setting new goals. Hence, you'll never be ready to move on to a new language.
•"I'll try to learn both English and the other language simultenously (assuming I don't need to start learning this new language from scratch). "
Yeah, why not... Just give it a whirl. You can always reconsider whenever you want.
•"The second approach would slow down my progress in English a little. Plus I would have to study two totally different accents (I still have to work on my English accent). And this is the main reason why I'm a bit wary of applying 2."
Yes, it probably would slow down your progress in English. But does that matter? Do you need your English to be at a certain level by a certain deadline? Maybe it doesn't matter if you become fluent in 3 years or 3.5 years? Only you know the answer to that question.
As for two different accents. Unless the languages are very similar I doubt there would be any interference between the two accents.
•"To put it simply: Is it effective to learn two foreign languages simultaneously? Are you for or against? "
Effective in what sense? Yes, it's equally effective. Simply the results of your efforts are distributed across two languages rather than one. Is that less effective in your view? Do you define effectiveness in a local or a global sense?
•"I'd rather speak 2 or 3 languages pretty well than 5 or more and end up scratching the surface. Perhaps this is why I wonder what would be most optimal in my case."
In my opinion 2 languages is optimal, 3 is doable in ideal conditions. 5 is a pipe dream suitable only for psychotic monomaniacs.
So, just start learning and decide for yourself. One month of lessened attention is not a death sentence for your English.
