This Has Always Bugged Me

Why couldn't she go driving alone? Elizabeth always went places alone and all of the Bennets were out in society before the elder were married. Jane Austen gives her characters a lot of independence.
I believe the second scene you're referring to is more Hollywoods way of telling us that Emma is a good person.
I can't give an answer to the last one

It was entirely appropriate for a young lady of Emma's standing to ride around her home estate in a barouche. You may be confusing this with the scandalous carriage scene from Sense and Sensibility where Marianne was frolicking and giggling with Willouby as they rushed through the village in his barouche. That was not appropriate because she was with a single man, unchaperoned, and her behavior lacked decorum.

#1 -- answered by others.

#2 -- the ladies of whichever "Great House" owned a particular estate absolutely did go around to visit the tenants and help them out whenever possible. It was most definitely a part of the job of the woman who ran the house. She would bring gifts of food and clothing (some of which was made by the ladies at the "Great House"), would listen to their troubles and would dispense advice.

One of the things that "Death Comes to Pemberley" did right was to have Elizabeth visit tenants on the estate and to help them out. Again, it was part of her job as "lady of the manor."

Good landowners felt a tremendous sense of responsibility for those who lived on their estates and paid a lot of attention to them and what went on in their lives.

#3 -- Absolutely. Women most definitely did shoot (arrows, not guns). There's even a book on the subject, called Archery, Romance and Elite Culture in England and Wales, c. 1780–1840. It may be found here: http://www.academia.edu/203053/Archery_Romance_and_Elite_Culture_in_England_and_Wales_c._1780-1840

Here is the abstract:

During the sixteenth and seventeenth centuries, the longbow was rendered redundant as a weapon of war by technological developments such as the musket. Yet at the end of the eighteenth century, archery was revived as a fashionable pastime amongst the English aristocracy thanks to a nostalgic taste for the gothic and medieval. Archery societies were set up across the country, each with its own strict entry criteria, outlandish costumes and extravagant dinners. In a period that saw the making of the modern British upper class,as landowners became more powerful, more unified and more status-conscious, archery societies were havens of exclusivity and a way of reinforcing and reassuring one’s own position in society. Furthermore, women could not only compete in the contests but retain and display their ‘feminine forms’ whilst doing so, and thus the clubs also acted as a forum for introductions, flirtation and romance. This article explores the meaning of archery for upper-class men and women and demonstrates how wider social needs and interests shaped play, recreation and fashion.

Throughout Austen's novels, young women out on their own, unaccompanied, are commonplace.
One of the accomplishments of young well-to-do women at that time was assisting the poor, in person.
Archery has long been a sport engaged in by women; however, there is no archery in Austen's novel "Emma".