Yes on both counts. Article 3 of Qatar's Commercial Companies Law (Law No. 11 of 2015) states clearly that every company incorporated in Qatar is considered a Qatari company, and its headquarters must be located in Qatar. You cannot incorporate a company in Qatar while basing its main operations or registered office abroad.
This is particularly relevant for expats who may be tempted to register in Qatar for licensing purposes while running the business primarily from another country. Doing so could create legal complications regarding the company's validity and regulatory compliance.
It is worth noting that Article 3 also includes a caveat: being classified as a Qatari company does not automatically entitle the company to all the rights or privileges that Qatari nationality confers — the law leaves room for specific provisions on this. Additionally, Article 17 confirms that foreign companies operating within Qatar (without being incorporated here) are still subject to Qatar's commercial companies law in terms of how they conduct business, though the rules around incorporation do not apply to them. If you are unsure whether to incorporate locally or operate as a branch of a foreign company, consult both a legal and commercial adviser familiar with Qatar's business environment.
This is general legal information, not legal advice. For advice on your specific situation, consult a lawyer licensed in Qatar.