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1) Ideally, some preparation should be done during the day and some right before immersion in the mikveh. If you prepare at home, you should quickly shower, comb through your hair, and check yourself for chatzitzot at the mikveh, just before you immerse.
There is no requirement to prepare at a specific time of day, as long as when you actually immerse in the mikveh you are free of chatzitzot. If your schedule demands that you do some things in the morning, that is fine. Durable preparations, like cutting your nails, can even be done a day or two in advance. The most important thing is that you look over your body (iyun) before immersion, and go into the mikveh clean.
2) The purpose of bathing before immersion is to clean yourself in general, and also to help remove scabs or at least soften them. Thirty minutes is often recommended, but is not a halachic requirement. A shorter bath is fine as long as it serves this purpose. If no bathtub is available, a thorough shower is also acceptable. There is no specific time by which the bath or shower should begin or end.
3) You can officially immerse in the mikveh after tzet hacochavim (when the stars come out, between 20 minutes and an hour after sunset, depending on geography and local custom). You need to find out what time that is where you live (it is generally around havdalah time). If you are in doubt, the mikveh attendant knows when to allow women to immerse.