1.) No, that is an appropriate amount of information. You want to give them WHO you are, WHERE you are, and WHAT you want. This gives them everything they need.
2.) The thing you have to know about VFR Flight Following/Traffic Advisories (the two terms are interchangeable) is that it is only a secondary function for the controllers. This means that if the controller gets too busy, you become a secondary priority. This is Note 2 of the definition of TRAFFIC ADVISORIES
from the FAA Pilot/Controller glossary:
Traffic advisory service will be provided to the extent possible depending on higher priority duties of the controller or other limitations; e.g., radar limitations, volume of traffic, frequency congestion, or controller workload. Radar/nonradar traffic advisories do not relieve the pilot of his/her responsibility to see and avoid other aircraft. Pilots are cautioned that there are many times when the controller is not able to give traffic advisories concerning all traffic in the aircraft's proximity; in other words, when a pilot requests or is receiving traffic advisories, he/she should not assume that all traffic will be issued.
The emphasis above is my own. Basically, even if you're under Flight Following, you need to keep a watch out for other traffic, just like you would if you were not under Flight Following. That service is only a tool to aid in seeing and avoiding other traffic.
What would have happened is that the next controller's sector would have probably observed that the first controller still had a "track" on your airplane, and would have called up the first controller to let him know to drop track and terminate your radar service, or to hand you off to the next sector. If not, then the next controller may or may not have recognized your squawk code as a code assigned to VFR aircraft and could have chosen to ignore it or to call up an adjacent sector to see what's up. Other means of contacting you would have ensued if the next controller did not recognize the squawk code, but you would not be in trouble for that as the pilot.
The important thing here is that you recognize that you still have to remain responsible for everything from traffic collision avoidance to avoiding Class B, C, and D airspace without permission; the radar controller providing you with Flight Following does not carry the responsibility of ensuring that you are given a frequency change. One of the worst case scenarios is that you enter another sectors's Class C airspace or something because you weren't appropriately handed off or terminated by the last controller. Yes, it is bad controlling technique to not hand you off, to not notify you of your encroachment on Class D, C, or B airspace, and even to not notify you of terrain and obstructions along your route of flight. But ultimately as a pilot you cannot make any assumptions about a controller's use of good controlling technique when he or she is providing you with a secondary service.
3.) Generally no, but you can and should advise ground that you will be seeking Flight Following after takeoff. If the local procedures allow for it, than the ground controller will assign a squawk code to you. If there are no procedures for that, then you're on your own.
Finally, what I recommend most is touring the ATC facilities off of which these questions are based. There's no better way to get a definitive answer (especially for your last question) than by setting up a tour with one of the facility supervisors and interacting with him/her and the controllers at the facility on your tour.
The phone number for the Houston TRACON is 281-230-8400. The phone number to Houston Center is 281-230-5600. You can find the number for your local tower at this page: http://www.stuckmic.com/texas-air-traffic-control-facilities.html
. If the number to the tower isn't there, then go here: http://www.stuckmic.com/texas-federal-contract-towers.html
and search for it.