Do you really pronounce -ой endings like that?
Yes. In fluent speech and this case, -о- becomes imprecise vowel, which reminds you "-ы-" (moreover, if spoken clearly and louldy it will be "-а-"). "-й" is spoken fluently also, but I can recognize it. Just "-ы" sounds not like this.

на этот период instead of в этот период (as it is written in the subtitles). Are both variants correct?
Yes.