Himself being a Brāhmaṇa, was it not wrong on Droṇa's part to learn warrior skills and impart them to Kṣatriyas?
No. It's not wrong on Droṇa's part. It is actually recommended that a brāhmaṇa learn several skills, not just warrior skills and teach them to others depending on their varṇa (caste). But he should never put those skills to use and earn income off of them. I suspect it's due to this reason, Droṇa asked the young graduates (Pāṇḍavas and Kauravas) to attack and capture Drupada when he could do it all by himself. Though he was quite capable of accomplishing the task himself, he didn't, because he shouldn't.
Swami Chandrasekarendra Saraswati in the book Hindu Dharma: The Universal Way of Life explains as below.
The Brahmin must be conversant with the fourteen branches of the Vedic lore. He must be proficient even in Gandharva-veda or music and must be acquainted with agricultural science, construction of houses, etc. At the same time he must give instructions in these subjects to pupils from the appropriate castes. His own vocation is the study of the Vedas and he must have no other source of income.
Viśvāmitra was the master of Dhanurveda (military science). When he performed sacrifices, the demons Subahu and Marica tried to play havoc with them. Though a great warrior himself, he did not try to drive away the demons himself. Instead, he brought Rāma and Lakṣmaṇa for the purpose. Viśvāmitra thereafter gave the instruction to the two in the use of astras and sastras.
If the Brahmin is asked, "Do you know to wield a knife? " he must be able to answer, "Yes, I know". If he is asked, "Do you know to draw and paint" again he must say, "Yes". But he cannot wield a knife or become an artist to earn his livelihood. All he can do is to learn these arts and teach others the same according to their caste. He is permitted to receive a dakṣiṇā to maintain himself and he must be contented with it however small the sum may be. The Brahmin's speciality, his true vocation, is Vedic learning.
[Hindu Dharma » Varna Dharma For Universal Well-Being » Brahmins are not a Privileged Caste]
Wasn't he breaking the rules of dharma when he actively participated in the Kurukṣetra war?
Yes and No.
Yes. Broke the rules. Because, as explained above, he cannot use the Dhanur-veda he learned in a real battle. He is only allowed to teach those skills.
No, didn't break any rules because he's obligated to fight for his hosts (Dhṛtarāṣṭra, Duryodhana, Bhīṣma etc.) for having given him a place to stay and lead a comfortable life. Even if it meant going against his sva-dharma to raise weapons against people.
However, if Droṇa made a case to Bhīṣma or Dhṛtarāṣṭra to relieve him from the war (Virāṭa or Kurukṣetra), my guess is they would be obliged to honor his request.
But with Droṇa not participating, I doubt if the Kurukṣetra war itself would have taken place!