Question: Does one violate a Biblical prohibition by using a non-toveled utensil? 

Short Answer: It is a machlokes whether one violates a Biblical prohibition by using a non-toveled utensil. Regardless, one may not delay toveling a utensil (that he uses), and if he does, he possibly violates a mitzvas asei (positive commandment).

Question: Is it permitted for a man and woman to have yichud where there is a video stream recording them?

 Short Answer: Most poskim prohibit yichud even where there is a camera videoing the seclusion. The poskim who do permit yichud only permit it under certain limited circumstances, as described herein.

Question: Is there a prohibition of yichud when a male convert is secluded with his mother?

 Short Answer: This is a major dispute amongst contemporary poskim. The vast majority, though, are lenient, as there still is a familial connection between the convert and his mother.

Question: May a man have yichud when his young son or daughter is sleeping in another room?

 Short Answer: A man may have yichud with a woman when his young child (ages 7-9) is awake in the house, even if the child is in a different room (as long as the doors to the rooms are open). However, yichud is forbidden if the child is asleep.

Question: May a man have yichud in a back room (e.g., a den) of a house when the front door to the house is open?

 Short Answer: Yichud is permitted in a back room if the front door to the house is open (or according to some poskim, merely unlocked) and the back room is open directly to the front room. There are some poskim, though, who might permit it even if the door to the back room is closed (but unlocked).