Good idea.  We have made other nitro-aromatics or -heteroaromatics and the carbon attached to the nitrogen is often broader than the others.

Generally Claisen condensation

Ester enolates can be formed with alkoxides, and they are good nucleophiles
Esters can undergo addition-elimination reactions with nucleophiles, otherwise known as nucleophilic acyl substitution. In the Claisen condensation the nucleophile is the ROC(O)CH2 group which displaces (RO-).
Alkoxides can be adequate leaving groups in this case, since they’re significantly weaker bases than ester enolates.
The product of a Claisen condensation is a beta-keto ester, which is a significantly stronger acid (pKa = 11) than either alcohols (pKa = 16-17) or esters (pKa = 25).
The reaction requires at least one equivalent of base since the enolate of the beta-keto ester product is not a strong enough base to deprotonate the starting ester.
A “crossed” Claisen condensation involves the enolate of one ester reacting as a nucleophile with a different (ideally non-enolizable) ester to give a beta-keto ester.

Mixed Claisen condensations are possible if the excess ester does not have protons at C-α. If both esters have α-protons, predominantly statistical mixtures of products can be expected.

We are short one signal in the C-13 spectrum; we could use LC-MS data to confirm this product.

No idea.

Note: in most cases when it comes to reagents it doesn't matter how they were produced - a compound is a compound, no matter where it does come from. Things like concentration, pH, presence of other reagents recipe calls for are much more important. In practice sometimes (rarely, but often enough it is always a thing to remember) minute amounts of byproducts (which do depend on the production method) can influence the output of a process. The purer the reagent, the lower chances of the problem - but purity typically comes at a hefty price.

As I wrote earlier - the only to make sure it works is to try.

You can always try to buy solid FeCl₃ from chemical suppliers, like Sigma-Aldrich, and make the 50% solution by yourself.

Trick is, devil may be in the purity - reagent grade ferric chloride is about 98% pure, and the other 2% are not well defined, yet they can be crucial for your application. The only sure way of checking it by trial and error.