For obligatory ghusl during Ramadan, does one have to do ghusl before the beginning of Fajr or before the end of Fajr?

It is Ok to make ghusl at any time but it is better and highly recommended to do it before Fajr so you can catch the Fajr Salah on time. But fasting is Ok even if someone would take ghusl after Fajr.

Ramadan is coming, and there is question that is circulating through our ladies about whether it is better to organize an 'iftar for friends and community members or is it better not to because some say it takes time away from `ibadat; not the obligatory prayers but the nawafil. 

 I would prefer to try to do both kinds of `ibadah as the Prophet Muhammad (peace and blessings be upon him) encouraged to offer 'iftar, and at the same time, he stated that the reward of nawafil in Ramadan is equal to fard in other days.

However, if the choice between the two is difficult, then the priority would differ from one person to another. If the person is in such a position that he or she is supposed to serve others, then the work that is beneficial for others is better than personal `ibadah. But for a normal person, `ibadah should take priority.

 I wanted to know what is the age a child must start praying from? From what age is a child obligated to fast? 

The age of accountability or takleef is when someone reaches puberty for both fasting and prayer. But we have the hadeeth of the Prophet (peace and blessings be upon him) which instructs us to train our children from an early age.

Therefore, the age of 7 is the start of training our children, the age of 10 of taking it seriously and the age of puberty is the age of accountability. Maybe the fasting training age would be different from the prayer one, but we know from the practice of the Prophet's Companions that they were training their kids for fasting earlier than the accountability age.

So we are supposed to start ahead of time and educate and train our children so that when the age of accountability comes, they are ready to undertake the fast with their own desire.

 Ramadan Mubarak. I want to make sure what time should one stop eating. When Fajr starts or 15 minutes before fajr starts? And if the time table says to break fast at some time but by looking at sky, we are not sure if it is exactly time should we wait or follow time table or adhan? 

There is a real Fajr time when the streak of line comes from the horizon. This is called al-fajr as-sadiq. This is the time when we are supposed to stop eating and drinking. Now, what happens with Ramadan calendars is that some time people put exact timings and some others put pre-fajr time for the purpose of extra caution or ihtiyat or if someone woke up on that time they can take only a little bit of food. In some countries they raise fajr athan just after fajr, like in Saudi Arabia. So it is Ok to continue till you hear the adhan. But in other countries where they make adhan according to their own fixed timing so are supposed to follow the chart of fajr times and not the adhan.

As for the second part of the question, in summer time that enough light remains even after sunset. So we are supposed to follow exact timings even there is a light in the horizon, because Prophet Muhammad (peace and blessings be upon him) instructed us to delay the suhur.

If I get my period during the day I am fasting, do I have to make up that day even if it was just before 'iftar time? 

Yes, you need to make up for that day. You should continue this day as fasting but it won't be counted as a valid day of fast.

 I want to know if I can swallow my saliva (in my mouth) during fasting? 

Of course you can. This is something natural that you cannot stop, but some scholars deem it makruh to intentionally gather a lot of saliva in one's mouth and then swallow it; but it does not break the fast.

 Can we perform `Umrah Qada' or Qada' to make up for past neglected prayers in our `ibadat after Tarawih or during `itikaf? 

You can make your own schedule by either dividing your missed prayer on daily basis or after Tarawih or in `itikaf time, whatever is easier of you.