While I'm not a 737 pilot, I do know about the MCP pitch and roll modes, so I can answer as follows:
LNAV - is for lateral navigation, i.e a roll mode, and obviously cannot be used for vertical manoeuvres.
VNAV - can only be used if the FMC has been fully programmed, and a vertical profile has been calculated. This would be the most common mode used by airlines, and is typically engaged at 400ft after takeoff. Depends on company SOP's etc.
LVL CHG - Is typically used when ATC instruct an A/C to change to a different flight level, (climb or descent). It will use the current A/C speed, or the speed set on the MCP as the reference speed for completing the level change.
CWS- Is a degraded auto-pilot system. It is possible to make manual inputs using the control wheel and when the required roll/attitude is obtained you can release the wheel and the AP will hold that attitude until you make another input, or engage a CMD AP. I'm not aware of any UK airlines that use CWS as part of their SOP's.
I hope that answers your questions.