Before addressing the specific questions, some background on how the autoflight system works:
It's assumed that the pilot presses TO/GA when beginning the takeoff roll. This will clear any MCP modes such as heading or approach or level change or vertical spped if they were still engaged, but in the case of LNAV and VNAV those will remain armed to engage at the appropriate time (if the pilot had armed them already by pressing the button on the MCP).
(Not part of the question, but just for completeness, LNAV will engage, if armed, at 50' AGL after takeoff and initially command a straight-ahead track, commanding a turn as necessary to meet the loaded departure procedure. If the initial track in the LEGS page doesn't align closely with the departure runway, LNAV can't be armed on the ground.)
If armed, VNAV will engage at 400' and command V2 + 20 until the acceleration altitude (typically 1000' above the runway), at which time it will command 230 knots or the maximum airspeed for the current flap configuration, if less. And as the flaps are retracted, that commanded speed will increase, eventually matching the FMC climb speed schedule - typically, 250 knots until 10,000', then the ECON climb speed in IAS until transitioning to the ECON mach number climb speed.
If VNAV was not armed, then the TOGA mode will remain engaged indefinitely; there is no fixed altitude at which it will change over to something else. Per se, TOGA commands wings level and V2+20. At autopilot engagement, that changes over to Level Change and the window opens at the current speed. And of course if the pilot selects level change, then the window opens and that - along with Heading Select (or LNAV, if already active) - is the mode you're in. Any of those can be selected at or above 400' RA.
And, if the vertical mode goes into Altitude acquire, then the window opens at the current speed. Same if Vertical Speed is selected, the window opens at the current speed.
So for the scenarios asked about in the first paragraph of the question,
- I've never seen "taking off in level change" done in the aircraft. I suppose you might be able to engage the mode on the runway (along with LNAV or Heading Select), but it's not something I've ever seen done so I can't tell you how it would actually work. (You certainly wouldn't have autothrottles setting takeoff thrust for you, for one thing.)
- If you take off with nothing armed and without pressing TOGA, you simply have no flight directors or other automation engaged, and the speed window will remain at V2 until it's changed.
- If you had Level Change engaged but then press TOGA, Level Change is gone & you have TOGA vertical guidance.
- If you have nothing armed and press TOGA at the start of takeoff, then TOGA vertical guidance will stay engaged until some other vertical mode engages (either VNAV at 400', altitude acquire, or by a pilot pushing a button).