Forming a dioxolan with levodopa
Before answering the question, you might find it helpful to know that people tend not to go about making the dioxolane, but instead make use of natural products such as safrole to buy in a precursor with the ring already formed. If you work at a University or a research institution, this shouldn't be an issue. If you don't, I strongly advise you against trying this chemistry, as your desired product bears significant resemblance to multiple illegal drugs. 



To answer specific parts of your question: 


  1) Does the nitrogen or the carboxyl group react with the methanal? Do I have to use a protecting group?


The reactivity of the amino acid portion depends on the ionisations.  

In both the protonated and deprotonated form, the carboxylic acid/carboxylate can't react with formaldehyde. 

Amines do react with formaldehyde in the presence of an acid catalyst (think about the Mannich reaction). Whether or not your substrate (amino acid) as drawn would react with formaldehyde likely depends upon whether the amine is protonated or not (think about the amino acid and whether or not it would exist in its zwitterionic form under the conditions given). 


  2) Is there a chance it reacts similar to a marquis reaction?


The mechanism of the Marquis reaction isn't fully elucidated, however it does give positive results for many alkaloid drugs 


As already stated, given the similarity of your compound to several illicit drugs, I wouldn't be at all surprised if it did give a positive result. 



Moving on to the actual chemistry you've proposed. 

The standard method for making 1,3-benzodioxole (and derivatives) is to take the catechol and treat it with CH2Br2 under basic conditions. 

This works better than formaldehyde (I'm not actually convinced the formaldehyde would work at all, but I'll presume you've researched the method) because rather than trying to expel water (which as you point out, you'd have to Dean-Stark to remove the water), you only expel halide ions, which don't interfere under the conditions. 

(As another side note, experimentally you also need to add a phase-transfer catalyst as the CH2Br2 isn't miscible with the base/aqueous layer).