Question about the structures and conditions of validity of the mixed-effect model In class we are told to be very careful when using mixed models and to use the simplest models, to avoid talking nonsense. I posted questions previously about this but I will come back to the wording of the question.
It is about a group of people followed for a treatment against depression: 146 people (Men and women), 8 times of measure for each subject (But not all subjects did all the visits). I have to answer about if treatment works better in one gender group compare to the other. My variables of interest are ScoreHamilton (Score used to assess depression state), GROUPE (Gender: male or female),TEMPS (Different times of visit),NUMERO (Subjects ID).
So here is how compute my model:
modMix <- lme(ScoreHamilton ~ TEMPS + GROUPE, random = ~ TEMPS | NUMERO, 
              data = Ham_norm.mix)

However, I have noticed that most of my classmates have used a model without a random slope: ~ 1 | NUMERO. In my opinion, this biases the results.
Maybe without knowing the structure in my dataset, you can't advise me properly.
However, what risk would I also take by using a random slope (which I find reasonable in this case) instead of a common slope?
I would like to know in practice what are the minimum conditions of validity (the most important: distribution of residues? Linearity of mean score trends?) to check when using a mixed model. Because I noticed the way to check the conditions of validity different from one person to another.
Is it very important to check the form of the correlation matrix to be used in the model (Toeplitz Matrix? MatrixAR(1))?
I know that lme4 is a newer package than nlme, but how could I know the significance of a coefficient with the latter if I want to use it?
 A: 
However, what risk would I also take by using a random slope (which I find reasonable in this case) instead of a common slope?

Provided that the model converges, and the estimates for the random effects are not unreasonable (ie. they are not close to zero) there isn't really a risk. Random slopes allow the effect of the variable to be different in each subject, and this is often a natural thing to expect. You can also test, with a likelihood ratio test which model fits better.

I would like to know in practice what are the minimum conditions of validity (the most important: distribution of residues? Linearity of mean score trends?) to check when using a mixed model. Because I noticed the way to check the conditions of validity different from one person to another.

If your objective is inference rather than prediction, then the main thing is for the residuals to be plausibly normally distributed. This includes the random effects.

Is it very important to check the form of the correlation matrix to be used in the model (Toeplitz Matrix? MatrixAR(1))?

I assume you are referring to a correlation structure among the observations for each subject, such as autocorrelation, which is handled in nlme with the corAR1 function which you specify in the model and should ideally be informed by domain knowledge. In many packages (e.g. lme4) it is not even an option. Of course, you can test the residuals of your base model for autocorrelation.

I know that lme4 is a newer package than nlme, but how could I know the significance of a coefficient with the latter if I want to use it?

For LMMs I think lmer provudes p-values if that's what you seek. You can also use package lmertest 
