Carry-over subjects are the ones that you failed to clear in that academic year. If you have carry-over subjects from the 2nd year, you will be able to attend 3rd-year MBBS classes but you will have to clear the supplementary examinations for the carry-over subjects. 

Yes, you can write 3rd-year examination in June after clearing 2nd-year supplementary examinations in June. The course work of 3rd year isn’t heavy there is only one major subject, i.e. PSM along with two minor subjects. So, you will be able to manage. 

Carry over means that you can attend 3rd year classes but you will have to take the supplementary examination for the 2nd year subjects, in which you failed, after 6 months. 

Now the supplementary examination usually takes place in June. If you pass the supplementary examination for the 2nd year subjects in June, you can appear in the 3rd year examination in December. If you fail to clear your carry over subjects on the second attempt, you won’t be eligible to take the 3rd year examination. Now clearing your 2nd year carry over subjects won’t be that difficult, since 3rd year has only one major subject, i.e. PSM along with two minor subjects. If you work hard, it is doable.