Answer 1:
The Family Division is part of the High Court where solicitors don't have a 'right of audience'. Legal representation has to be through a barrister.
Answer 2:
As the person petitioning for a contact order, you do have a right of audience before a division of the High Court so, theoretically at least, you can 'do it yourself'. However, your chances of success are likely to be considerably diminished without the services of a barrister. (The main part of a barrister's work is often in the preparation of the papers prior to the hearing, rather than in the hearing itself).
In general, the law doesn't grant any rights to grandparents. If you've managed to get part way through your quest for a contact order, there must be some exceptional circumstances but it's clear that the case must be far from straightforward. If there's any way in which you can engage the services of a barrister, I strongly recommend that you should do so.
Perhaps the Grandparents' Association can offer some advice?
http://www.grandparents-association.org.uk/
Good luck!
Chris