Doubt regarding mixed modeling format Say, I have a dataset that looks at how many times my 5 babies chases a cat around the house . I'm trying to estimate 'y' which is the number of times the cat runs one complete round around the house as a function of the color of the dress the babies wear and the speed at which the babies are walking. But, I also have information about the type of food the babies ate and if the weather is cloudy or sunny on that specific day. What food the baby chooses to eat can be dependent on the weather.
I want to estimate y as fn of color and speed:
Now, my model is 'y'~color*speed+(1|baby)
But, I'm wondering if I can also add information about the food and weather, which are not part of my fixed effects into my model( without including interactions)
'y'~color*speed+(food||baby)+(food||weather)
 A: 
But, I'm wondering if I can also add information about the food and weather, which are not part of my fixed effects into my model( without including interactions)

Yes, you can. By not including them as fixed effects, but including them as random slopes, you are saying that the overall mean slope is zero, but each individual subject (baby and weather in your model) will have it's own slope. Whether it makes sense in your modelling context, is another matter altogether.
Note that in your 2nd model you include weather as a grouping variable for random intercepts. You said that there are only 2 levels of weather - sunny or cloudy - so in this case that would not make sense becuase the software will try to estimate a variance for a normally distributed variable based on only 2 observations. So in this case you would specify weather as a fixed effect.
Also note that in your 2nd model the || syntax means, at least in the lme4 package, that the software will not estimate a correlation between the random slopes and the random intercepts
