On your first query, the formula that you have above:
does not include a main effect for ‘treat’, that is a " … + treat + … " term, which may have been a typo.
To Frank’s point, under the principle of hierarchy, if you are including an interaction term in the model, you should also include the main effects for the terms involved in the interaction.
Thus, Frank’s formula, which includes
time + y0 + treat + time:y0 + time:treat
thus, taking advantage of R’s simplified syntax for model formulae.
Note that Frank’s formula does not include an interaction between y0 and treat, which I would include.
If you wanted to include all main effects and their second order interactions, you could write the formula as:
y ~ (time + y0 + treat) ^ 2 + (1 | id)
which would expand to:
y ~ time + y0 + treat + time:y0 + time:treat + y0:treat + (1 | id)
I would honestly be less concerned with the easier interpretation of the coefficients in this setting, and more focused on having the correct model specification, given the study design, sufficiency of data, and any other considerations.
One can then generate predictions and relevant contrasts from the model to assess treatment and time effects, and other relevant parameters. In R, for example, I use Russ Lenth’s ‘emmeans’ package to generate various relevant contrasts and simultaneous confidence intervals, with corrections for multiple testing as apropos.
On your second question, given the time since the original discussion on your post last June, and some of the confusion since then, it may be easier to just re-post the initial formula that I use in this setting, with the proviso that this is used in the setting where you have a baseline measurement, and there are at least two post-baseline time points and measurements. I am modifying it here for consistency with the lme4/lmer syntax used above, as opposed to the earlier lme syntax that I originally posted:
y ~ treat + time + treat:time + y0 + treat:y0 + (1 | id)
So I do have the “group * y0” interaction term, but do not have a “time * y0” interaction term. However, as I noted above, you could include both, and assess as apropos, if that makes sense.
The above, using “*”, would simplify to:
y ~ treat * time + y0 * treat + (1 | id)