Lawrence's red VW bug is visible in the parking lot of the diner and out front of Wendy's house. We don't see him get out of it in either scene so there is no way to know it is his.
This episode was originally planned as the second episode of the series -- a follow-up to Timmy getting cast in "Man of LaMancha" in the pilot --but the network decided that they preferred Peggy-centric stories and this episode was largely about Timmy's relationship with his dad, so this story got pushed back to number five. Originally there were several scenes of hookers from Hollywood Boulevard attending Timmy's play, but the bosses at ABC also asked that they be edited out.
Peggy smacking Timmy's head with a wooden spoon in this episode became a big issue with the standards department at ABC. The concern was that it constituted abusive behavior and that Peggy came across as too mean. After much debate with the showrunner the spoon scene and a later scene of Peggy comforting Timmy over his fractured relationship with Mike were both reshot with the intention of "softening" Peggy's demeanor. Also the sound effect of the spoon hitting Timmy's head was replace in an effort to make the hit seem less violent.