Answering these two first:

Particularly, what do the negative values mean? What does it mean to have a negative influence on accurately predicting the class?

If you look at the definition of how the partial plot is computed in the Random Forest package documentation, it shows that the plots show the relative logit contribution of the variable on the class probability. In other words negative values (in the y-axis) mean that the positive class is less likely for that value of the independent variable (x-axis). Similarly positive values mean that the positive class is more likely for that value of the independent variable. Clearly, zero implies no average impact on class probability.

And what is the most important feature from these figures, is it the max value, the shape of the trend etc?

There are many different approaches to determine feature importance and max absolute value is just one simple measure. Typically, people look at the shape of the partial plots to gather understanding about what the model is suggesting about the relationship from variables to class labels.

Can you compare the partial plots with partial plots of other variables?

The answer to this is less black and white. You can certain look at the range of the y-axis for each plot; If the partial dependence on one variable is near zero for the whole range of the variable, that tells you that the model does not have any relationship from the variable to the class label. Back to your question, the larger the range, the stronger the influence overall so in this sense they can be compared.

I have no experience with Maxent.

Answering these two first:

Particularly, what do the negative values mean? What does it mean to have a negative influence on accurately predicting the class?

If you look at the definition of how the partial plot is computed in the Random Forest package documentation, is says that the plots show the relative logit contribution of the variable on the class probability from the perspective of the model. In other words negative values (in the y-axis) mean that the positive class is less likely for that value of the independent variable (x-axis) according to the model. Similarly positive values mean that the positive class is more likely for that value of the independent variable according to the model. Clearly, zero implies no average impact on class probability according to the model.

And what is the most important feature from these figures, is it the max value, the shape of the trend etc?

There are many different approaches to determine feature importance and max absolute value is just one simple measure. Typically, people look at the shape of the partial plots to gather understanding about what the model is suggesting about the relationship from variables to class labels.

Can you compare the partial plots with partial plots of other variables?

The answer to this is less black and white. You can certain look at the range of the y-axis for each plot; If the partial dependence on one variable is near zero for the whole range of the variable, that tells you that the model does not have any relationship from the variable to the class label. Back to your question, the larger the range, the stronger the influence overall so in this sense they can be compared.

I have no experience with Maxent.