Khaled,
Your situation is not that odd, especially for a specialist just starting-up and trying to minimize overhead. Nonetheless, I think the advice you are getting here points out that sometimes trying to keep it inexpensive and simple may actually make lead to complex and expensive solutions....
In answer to your specific question, "what happens if two people are trying to access the server remotely with logmein?"...the first one gets bumped off. I don't know of any way around this.
How will you handle incoming phone calls when you are in the office? I have NEVER heard of a landlord who won't let you put in a phone line (and of course, a DSL is really the same).
Not to beat a dead horse, but you can have your server "locked" so no one on-site can access it when you are not there. "Taking it home" is NOT an advantage in your situation, because others need to access it. The database must be accessible to your staff, so I would keep it in the same spot (the office where you see the patients).
If you must do it the way you propose, some of the others on the board with more networking know-how than I possess may be able to help you.
