Is SAT compulsory for UIUC?

Universities usually do not give minimum SAT cutoff for admissions. So applicants should check the previous year's trends. Based on those scores, prepare and set a goal.

You should start preparing for SAT 3-4 months before your actual SAT exam date. This time is suggested so that you can cover SAT exam syllabus and also take practice tests. There may be few who would require more prep time. In this case they should plan their schedules as per their exam dates. SAT preparation months will give applicants understanding of exam, how it is structured and how to attempt it as well.

SAT Preparation in a month may be possible for some but may not be for others. The previous knowledge if appropriate then its possible. If the student has a strong background in any of the two sections of SAT or in both sections of SAT, they can easily prepare for SAT in a month because they just have to brush up their skills. While others with zero knowledge will not be able to do so.

No, there is no common answer on SAT. Each answer choice whether A, B, C, or D are evenly distributed, so, guessing a particular choice (e.g. option B) for most questions will not be the right approach. Even while preparing for SAT exam, students may have come across a couple of questions which may have a common option as answer but this rule may not apply at all times, especially for the main exam day. 

Heidelberg University provide test option to international students for ACT or SAT score. Student still choose to submit their score to boost their admission chances. Students should aim to score a minimum ACT score is 21 or above and minimum SAT score of 560 or above.