Save the Dates are Traditionally sent 6-8 months before the wedding unless it is a destination wedding and then they can be sent as far as 12 months in advance. Invitations are traditionally sent out 6-8 weeks before the wedding but modern etiquette has changed and more typically invitations are mailed 8-10 weeks in advance.
There is no hard and fast rule here that is largely dependent on the hosts wedding. Typically, all single adult guests should be invited with a guest. But with more intimate venues and tight budgets, this is not always feasible for all guests. More typically, we see plus ones reversed for couples with more serious relationships. The single guests who may not know anyone else at the wedding and could feel awkward attending solo.
This should be one month before the wedding. if your venue requires a final count of more than one month in advance then 5 weeks is also acceptable. Remember most guest will reply prior to the listed RSVP date. This date is the last possible day to reply and generally, it will only apply to a few stragglers.
Your indication or inviting children or not is indicated on the addressed envelopes. If your child's name is not included on the front of the envelope, then they are not invited to the event. Most guests are aware of this convention. However, some couples do opt for the subtle place to add adults attending on the rsvp card ie ___ adults attending while this is not traditional it is a respectful way of just reiterating that adults only should attend and we do find that for some clients this puts them more at ease.
Definitely.... with the invitation, you are indicating you would like for them to be there ( whether or not they can attend). In fact, if guests reply to a save the date to say they will not attend, you should absolutely still send them an invitation. If you would like to add a personal note that you are sorry they can't attend that is a nice touch. But if you don't send an invitation that indicates that they are no longer invited to the event.