You can file a small claims action if the claim is under the jurisdictional limit of $7,500 and the court has subject matter jurisdiction over the case. Claims for breach of contract will fall under small claims jurisdiction. Please see the court's small claims instructions for more information. Before taking legal action, I would suggest reviewing our article on unlicensed work, particularly the section on "Who is Responsible for Pulling Permits?"
Also, you should review your contract for an arbitration clause. This could stipulate how and where you may take action against the contractor. In the event of such a clause in your contract, the contractor could require you to pursue them through the arbitration process, which can be significantly more expensive.
You are also entitled to litigate the party with whom you first signed the contract, in this case, the former company. The question of litigating the new company is a more intricate issue. Companies can acquire other businesses through an equity sale or an asset sale. In an equity sale, the former company sells all or a portion of their membership interest (also referred to as stock in an LLC) to the new company. On the other hand, an asset sale involves the former company selling its assets, which could include your contract, to the new company. Thus, determining whether the new company is a suitable party to the lawsuit is something I can't definitively answer based on the information provided in this Q&A.
Pertaining to the sale, I suggest scrutinizing your contract for an anti-assignment clause, which would prevent the former company from transferring your contract to the new one. If this clause exists, you might have additional legal options at your disposal. Conversely, if it's not included, the general assumption is that the contract can be assigned
As for your last question, you have correctly identified the damages in your case. The cost to get the work properly permitted, assuming that the contractor has the responsibility for permitting the furnace and A/C unit.
Finally, I wanted to point out that depending on the claims brought, such as poor workmanship, Colorado’s Construction Defect Action Reform Act (CDARA) may apply. For a complete discussion see our article titled "Dealing with a Difficult Contractor: What Kind of Lawyer Do You Need," particularly the section on Colorado’s Construction Defect Action Reform Act.