This is a very natural question, but as it happens one needs some more background to give a natural answer (is my humble opinion). 

For clarity let me give a summary first indication:

As to your **Question 1.** : This is commonly thought of as not being possible in strict terms, because '*constructive mathematics*' is usually interpreted as '*that part of alternative constructive mathematical theories which is compatible with classical mathematics*.' 

Broadly speaking, this interpretation of 'constructive mathematics' is known as BISH (from Bishop-style mathematics). BISH is seen as the common core of classical math (CLASS), 
intuitionistic mathematics (INT) and 'Russian' recursive mathematics a la Markov (RUSS). 

Now if we include INT and RUSS in your term 'constructivism'. then the answer to question 1. is yes.

This because both INT and RUSS have additional axioms (compared to BISH) that are false classically. You could equally say that CLASS has an additional axiom compared to BISH (namely excluded middle) that is provably false in INT and RUSS. 

Famous examples of conflicting theorems are:

a) All total functions from $\mathbb{R}$ to $\mathbb{R}$ are continuous (a theorem both in INT and RUSS)

b) The unit interval is compact in the Heine-Borel sense (a theorem both in CLASS and INT, but false in RUSS).

-----

I do not understand your second question, could you perhaps elucidate a bit in the light of the above answer to question 1.?