Generally, we try to submit each question individually, but I'll do my best to answer these briefly from the Theravada tradition and give links for further information.
A lay Buddhist follows the teachings of the Buddha and observes the Five Precepts. A lay Buddhist also Takes Refuge in the Buddha, Dhamma, and Sangha.
The Buddha defined 4 types of followers: monks, nuns, lay men and lay women. The special name for a lay man is Upฤsaka and for a lay woman is Upฤsikฤ.
All four types of followers listed above are important. Buddhism was the first religion to accept women as equals. A woman can become enlightened. A Buddha however is always a man. However, with rebirths, we'll all be men, women, etc. at one time or another, so nothing needs to be ruled out. ;-)
As a practical matter, a woman in the Theravada tradition needs to keep some distance from an ordained male and a man needs to keep some distance from an ordained female. There is no physical contact such as a handshake with an ordained person of the opposite gender. Really the most appropriate way to greet a monastic is with palms together at the heart or at the forehead.
You can handle money, only monastics have rules against that.
On ordinary days you eat as normal, though mindful eating is always good. On Uposatha Days (or during a meditation course or retreat) you may wish to observe the 8 Precepts instead of 5 Precepts. In that case you would eat only between dawn and noontime. And yes, no alcohol at all.
You don't need a shrine. Some people like to have a statue of the Buddha or an area dedicated to meditation. But it's optional.
Yes, you can have entertainment although if you are observing the 8 Precepts you would avoid entertainment during 8 Precept observation. Buddhists who are concentrating on beginning or continuing a meditation practice often limit entertainment as it's not conducive to meditation.
If you observe the 8 Precepts you wouldn't eat after noontime. But you would generally eat a full meal before noontime. So extended fasting is not really part of Buddhist tradition. It's considered an ascetic practice and Buddhism is more about the middle way.
Ordained Buddhists shave their heads. It's not necessary for lay followers; but not disallowed either. The only clothing that is different for lay followers is sometimes at intensive meditation courses or on Uposatha days or other special occasions, lay followers wear all white. It's a tradition in Thailand and other Buddhist countries in Asia and sometimes that tradition is followed in the West as well. But not always.
A regular meditation practice is considered very important. Not all Buddhists chant, but that too can be considered an important part of the practice. Some people feel chanting helps settle their mind for meditation.
Yes, you can meditate at home and find support for your practice online in communities such as this and others. But the funny thing is, there can be Buddhist temples tucked away in lots of Western communities that people don't even know about. Many are smaller and don't really stand out. Some don't have websites or do any advertising or promotion. So it can take a while to know for sure whether or not there are any local Buddhist temples, groups, or meditation centers in an area. If there is a University near you, you might find a Buddhist group that meets there. I hope you'll be happily surprised to find something near you to support your practice. But if not, the online Buddhist community is a great help as well. Best wishes. :)