Hi! I think "He accepted the invitation and is looking forward to the visit." sounds the most natural. Hope this helped!

I think these two are ok: "He accepted the invitation, looking forward to the visit" (not 100% sure on this one) and "He accepted the invitation and looked forward to the visit"

If it's connected to the present you could say: 1) He has accepted the invitation and is looking forward to the visit. If it's strictly in the past (finished actions): 2) He accepted the invitation and was looking forward to the visit. We tend to say 'looking forward to' in a more informal way and 'look forward to' in a formal situation, such as at the end of a business letter/email/job application. Example: I look forward to hearing from you (at your earliest convenience). So it's really a matter of formality.