I am not familiar with that vehicle. These types of issues often follow a specific model. Best route if you want to work on it yourself is find a community. That car has a following, so find a forum of mitsu/laser owners.
Other than that, the trans problem sounds like a logic issue. In general, it's a hydraulic system of clutches and gears and in yours likely controlled by an electronic system of pressure transducers. If any of these components start to fail it can cause weird engagement problems.
If the problem didn't come and go it could be clutch material. (Especially with the neutral burnouts.) Depends on mileage. If the problem only happened the first time each day, especially on cold days, then you could just have a bad check valve. Since the neutral back & forth fixes it I'd say either you're giving enough time for the valve body to fill (after a check valve allowed it to drain) or there's a failing electronic component.
Really, you need to find people who know that car and its idiosyncrasies.
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