I've never seen or heard the explanatory "that is" shortened to "that's", but it wouldn't hurt to be sure if it's possible or not
When you recount a story, you can either use past or present tenses, but consistency is key. That is, if you choose present tenses, for instance, to portray some of the events of a story set in the past, then you should stick with present tenses for the entire story.
thus
instead.so
instead?