In 1 Cor. 11:1-16, do angels learn from watching men and women and do they pay attention to things such as dress? Also, in 1 Sam. 16 we learn that God looks on the heart: do angels have the same ability?
Living Stones CC
The apostle Paul is setting out in this portion the truth of Headship, that God has established an order (v.3): the Head of Christ is God, Christ is the head of man (as opposed to woman), and man is the head of woman. Headship brings in direction and nourishment and we see how our blessed Lord fulfilled, and continues to fulfill, as Man, His role as the Servant of Jehovah. He perfectly completed the will of God in every detail of His life and now we, both men and women in our respective roles, are to be governed by headship. Whenever a believer is praying or prophesying, direction (the outflow of headship) as to our head covering has been given by God, and we are to follow that direction. Angels are the servants of God, but are not in relation to God as believers, but they learn from our behavior (see Eph. 3 and 1 Cor. 4). In summary, our behavior is observed by angels, as well as other persons, and so our dress and words become a reflection of our profession. In all of our profession we are to be modest and our deportment is to be such as to bring honor to the name of our Lord Jesus. Note that these instructions as to head covering are given in relation to individual behavior, and not, as some may limit, to assembly functions.
As to your second question, only God looks on the heart. Angels and men look on the outward appearance. Angels are ministering spirits sent forth by and receiving instruction from God as to their work. Angels are not to be worshipped nor will they rule in the world to come. They desire to look into the things testified beforehand (see 1 Pet. 1), and there are myriads of them who are mighty, holy, and elect. There are also those fallen angels who are reserved for judgment.