Is there also a leveling effect for weak acids in water?
To answer your first question, weak acids don't get levelled in water. Each solvent has their acid-base equilibrium window, and water has one that is between 0-14, meaning that any solvent with a pKa within that range can be distinguished in water. There are other solvents with a ore narrow acid-base discrimination Window, such as methanol can acid, with a range between -10 to about 3, which will be unable to discriminate between weak acids. The reasoning behind the acid-base discrimination window is a bit long winded (and maybe a little confusing), so I will try to break it down as much as I can. 

Say you have an acid in water, then the pKa + pKb = pKw 
This is true for any acid or base in water. If pKa&lt;0 or pKb&lt;0 then then most of the compound will be dissociated in water, as an acid or base, respectively. This is undesirable, especially if the pKa/pKb&lt;&lt;0. As you have noted in your comments, when the pKa of a compound is less than 2 units compared to water, then in water only 1% of it will remain in the undissociated form. If pKa>pKw, you will obtain a negative pKb which will behave more like a base in water. It follows that compounds with a 0&lt; pKa&lt; pKw  will fulfill the condition where both pKa and pKb> 0. Now guess what pKw is? 14. Hopefully this might explain to you why water is able to distinguish between weak acids with a pKa between 0 to 14.