My recommendation: Apply now, October might be too late if you set the January 2019 deadline for yourself.
I grouped the considerations that led me to that answer in the three buckets General / Consulting companies / your readiness.
1) General: Forget December and the first half of January for interviews. Everyone's life (including yours) is too busy to finish what they promised until the end of the year. In addition: getting gifts, all the family affairs, travelling, and typically most people are taking a few days off over the Xmas/New Year's period. Then the usual lag until business restarts at the beginning of January: many people are away for the first two weeks, weather complications, etc.
2) Consulting companies: Sometimes it takes 1 month or more to get an answer after you applied. Don't underestimate that delay. If you apply by October, you might not get an answer before November, and the interview slots for November are taken by then. Furthermore, the Cons companies I know in Switzerland/Germany don't do a recruitment day in Dec, and I assume that mostly true for other locations, as people are equally busy to finish off client work, hand over delivarables, prepare the start in the new year, etc. Hence an application in October might not get you an interview before mid-/end-January.
3) You want to get your offer probably in/before November to start in January, rather than to worry over Xmas about your pending interviews. Not ready for the case interviews yet? No problem, should you get invited after you apply now, try to get an interview date later, e.g. in October/November. Gives you plenty of time for preparation, and you have a chance to get the job offer before the critial December period.
To your second question: Depends. Usually, you can start in any month of the year, usually on the first of each month. Sometimes people start mid-month (around the 15th), too, but that's less common. Some companies have special induction camps for their recent graduates, and they might have these camps 3 times a year. Hence, they might conduct interviews throughout the year, but effective start dates are only e.g. January, May, and September.