Those sentences sound unnecessarily convoluted to me, and "would" doesn't seem to be a good choice. "would" in such real situations as in your examples is only used when you're not sure about something ("I would say it is ..."), or to soften the tone when making a request ("I would like to ask you to ..."). In my opinion, your examples don't fall under these categories and should use the more clear and definite "will". I think most people will say those in shorter phrases in the active voice and not using "would". For example: 1. As I mentioned, I will update you (on [the subject]) by email after this chat session. 2. Rest assured this will be looked into and will get fixed. (even better: Rest assured we will look into this and fix it)

I don't know the name of the tense, but it's the same as will be updated. It's a strange phrasing to my ear unless it's like, "He told me it would be updated tonight" vs "He told me it will be updated tonight". 1st sentence is less affirmative. You'd never say to another person "I would update it tonight" without following it with a but or if -- I would update it tonight, but I have band practice...