Has the college started for 1st year B.E? If not, when can we expect it to start?

SJCE Mysore does not offer a 1st PUC (Pre-University course) program.

Yes, But consider this also-

local placement is decent, teaching is decent, overall exposure/brand is Not as Esteem-able as BMSCE/MSRIT/Amrita.

For vibrant city life, internships, coding culture, and good placements, MSRIT or BMSCE Bangalore CSE is an excellent choice.
  
In relation to the highest packages and national ranking, Amrita Coimbatore CSE is great if you don't mind a peaceful disciplined campus.

NIE/JSS Mysore are decent options if you don't mind Percentages and comparing to the numbers of 'big company' placements to some of the Bangalore Colleges above. 

Yes, Sri Jayachamarajendra College of Engineering offers two-year MCA programme divided into four semesters. The course is offered at the PG level. Candidates who have completed three-year bachelor's degree and Class 12 with Mathematics with at least 50% aggregate are eligible to apply for the MCA programme.

There are a total of 580 seats in the BE programme of SJCE Mysore. Candidates are offered seat in BE based on KCET or COMEDK scores. Below is specialisation-wise intake for BE:

Sri Jayachamarajendra College of Engineering offers four-year BE programme divided into eight semesters. Candidates need to pass PUC or Class 12 in relevant subjects to apply for BE. Below are some of the BE specialisations offered at SJCE:

• Civil Engineering
• Computer Science and Engineering
• Electrical and Electronics Engineering
• Mechanical Engineering
• Environmental Engineering, etc.