so, here are the phrases and my best attempts at how they want I should say this in English. Corrections wanted.
"If you want to go, so you should get your ticket now" -> If you want to go, you should get your ticket now." "So" is, I am informed by Sarah Benor, the Yinglish.
"I am staying by my sister for shabbos" -> ??? "I am staying at my sister's home/house/apartment over shabbos". Or "on shabbos"? "For shabbos" makes the most sense but I am second guessing.
"When I said that, I was going on the textbook, not the lecture" -> "When I said that, I was relying on what was written in the textbook, not said during the lecture"
"Dinner, have you had?" -> "Have you had dinner yet?"
"Do you have what to bring?" -> "Do you have what you need to bring" or "Do you have anything to bring", I THINK. However, my feeling is "what to bring" and "what to eat" are not SPECIFICALLY Yinglish and others do that too.
"I am getting also salad, and also chicken" -> "I am getting salad and chicken", although I feel English speakers are expected to reverse it, chicken precedes salad, so "I am getting chicken and salad" or even "I am getting salad with chicken" depending on what dish is being referred to.
"Do you have what to eat?" -> see above, "do you have anything to eat".
"I didn't think I would need, so I didn't bring" -> "I didn't think I would need it/the referrant, so I didn't bring it."
"I want that he should come" -> "I want him to come."
"Do you want for me to come with you?" -> "Do you want me to come with you."
"I felt badly for him" -> per @wenevergotusedtoegypt, this is "I felt bad for him."
"I will eat by them tomorrow" -> "I will eat at their house/apartment/domicile tomorrow"