I would not cater to your husband's fears; they are his and acting as if they are "true" will only hurt the relationship and him? Claim your share of computer time, at least equal to his time on the computer and don't "hide" anything but also don't offer to share if it is not an interest of his, if he is only asking because of his own fears.

If you pick "off" times to use the computer it will look like you do have something you are ashamed of or wish to hide? Maybe post the chat schedule if you go to planned chats or mark when you are going to check in because you told an online friend you'd check in that time to "meet" them. The more straightforward and "natural" you can make your actions look, the more they are important to you and you act like you are doing what interests you, the less anxious he should become over time perhaps.

Explain things as well as you can once and then refuse to explain further; my husband and I have a "code" for when I'm telling him about something he doesn't "get" (fantasy :-) and I tell him, "It's one of those things you don't get" and we drop it. Sometimes, he'll ask, "Is this something I'll understand?" and I can think about what I'm saying and realize there's really no purpose to discuss it, it's a "Perna thing" and not something of interest to him or something I could ever explain. We all have our interests and beliefs, etc. and you yourself know you are not cheating so his jealousy or worries are just that, his. You can be sorry he feels or worries in that way but it's not like you can fix him, only he can figure himself out and work through what he wants to do about his situation.