Flight Following into Class C Airport where I'm Landing
The answer to this depends a bit on what happens in practice as in my experience flight following does not always orchestrate handoffs as well as say an IFR flight would.
In some cases FF may drop you before you get to the class C airspace and you will have to request clearance into the airspace as if you were flying without flight following.
They may hand you off to the approach controller, and you can always ask for a handoff but FF is under no obligation to provide that. Remember flight following is NOT an airspace clearance and you will need to obtain clearance as you normally would (for class C thats positive radio contact that does not include a diversion or hold instruction). They will generally hand you off with time/distance to deal with this but you should have that approach frequency queued up in the radio.
Yes you should always report your altitude and position (when appropriate) and positive receiving of the weather when calling an approach controller for entry to their airspace assume FF has given them no information. You could say:

So Cal Approach Cessna 1234R at three thousand feet, inbound for a full stop landing at KLAX with information X-Ray

You should report position if you are not handed off and/or you feel as though local procedures require it (maybe FF is provided by a non radar entity), you would add something like

5 miles south of KLAX


Once on the ground you can ask ground, or the tower before takeoff for a flight following handoff. In my experience they will orchestrate this if its a quiet day and they dont have a lot else going on, other than that they are likely to tell you to contact approach in the air and request it.
They dont really need to know anything beyond Cessna 172 which gives them the performance information they need.