Confusion between the de Broglie wavelength of a particle and wave packets

I think it would help you to study the theory of Fourier transforms! Then this momentum/position duality becomes much more apparent.

A wavepacket is, like you write, a sum of many momentum states (not probability densities). If you look in the momentum space, the wider the spread of the momentum states, the smaller the spread of the position states will be. This is quite obvious from the Fourier theory which is why I recommend studying this. The Heisenberg uncertainty relation of position/momentum is related to this duality - when you make the momentum spread thinner, the position spread will increase and vice versa so there is a non-zero minimum of uncertainty (width of the distribution) where you have made both position and momentum as localized as you can.

The de Broglie relations simply relate momentum p to wavelength lambda - actually not more interesting than the simple observation that a sinusodial wave has a frequency and a wavelength. Short wavelength means higher momentum. So if you have a state with an uncertainty in momentum you also have an uncertainty in the de Broglie wavelength. If you want to take an average of it, go ahead as long as you know it's an average of a distribution of a certain width. This will be OK for many applications, but for some the detailed spread will be crucial also of course.

Addendum: Also please note that the position and momentum descriptions are a duality. You cannot specify both, the full information of the state is in either one of them and then you can Fourier-transform between them to extract a better understanding of the problem or extract some numerical predictions etc. This point is lost in some introductions so I'll take the opportunity here to mention it :)