Hello laurel2;
1. I know I can set up my own name-branded site -- am wondering if we are able to include domain checking and puchasing on the site?
To provide domain names you would need to become a reseller with a domain registrar. To be a domain reseller it is like $99 a month + most take a percentage of each domain registration. (I only know of two companys who offer domain reseller accounts) The reason for this is getting a license to be a top-level domain registrar is extremely expensive. Last I have heard it was several thousand dollars just to get considered and your consideration isnt 100% guaranteed.
2. Do future customers pay LunarPages sign up via my site and payment is made directly to LunarPages? At time of renewal, how is the customer notified and where is payment received.
The payment would be to you; when you purchase reseller hosting you will be provided your allocated space / bandwidth, and have a webhost manager to control it with. Once you reach your resource limit you must purchase addon space / bandwidth if you wish to keep expanding. (I would suggest a dedicated server if you plan on hosting for a profit.
3. I understand each customer will have their own control panel for website and email maintenance -- if there are problems or questions, do they come to me and I go to you?
They will come to you, unless you send them to Lunarpages. (I am not sure if lunarpages allows this or not.)
Keep in mind, I am not staff so this is not set in stone until a staff member replies.
Hope this helps.