Question: Did Stu suspect anything was wrong? Mrs Doubtfire insulted him saying his car was to compensate for his manhood and threw a lime at the back of his head, and he still invited her to the birthday dinner he booked? Was he not at all suspicious?
Answer: My interpretation is that Stu was temporarily ignoring/dismissing Mrs. Doubtfire's behaviour, as his relationship with Miranda was still new. After he married Miranda, he would be in a better position to complain to her. Or, he might think that Mrs. Doubtfire is defensive of the family, and he hopes to earn her approval later on.
Answer: Stu may have bypassed the comment as it only looked like it may have bothered him for a few seconds, but he got past it quickly. As to the lime, Mrs. Doubtfire directly told him it was an angry member of the kitchen staff who threw it, so he wouldn't have got angry at her for that incident.
Question: At the swimming pool, Stu goes to the bar and starts speaking to someone he knows. Was Daniel not the least bit worried that Stu's friend would say something and blow his cover? He would've seen "Mrs Doubtfire" lob the fruit, and then "she" even spoke to Stu's friend in a man's voice saying, "What are you looking at?"
Answer: It doesn't appear that Stu's friend was too familiar with the Hillards personally. He first asked whose children those were (as if he was unfamiliar with the kids), and although he was familiar with the name "Miranda Hillard", he may only know her as a mere acquaintance or through his job somehow (but not on a personal level). There is no indication he knew Daniel or had met him before. Also, he just saw Mrs. Doubtfire sitting there and just figured she was a random member, so he didn't think twice about her—regardless of the lime throwing or comments. Additionally, when he mentioned the kids' real father, it doesn't mean he knew Daniel; it just indicated that the kids seemed happy, and he figured they had a traditional two-parent background, so he asked about the "father" after Stu mentioned they needed a father figure in their lives.
Answer: With the comment about the Mercedes badge, I think he just thought she was being a bit too blunt and direct. The comment she made is actually quite common and is still used today as to "insult" someone. When she threw the lemon at his head, I think he knew it was her due to all the laughing. When she said it was an angry member of the kitchen staff, Stu didn't alter his gaze and look for the member of staff so I think he knew or was at least very convinced it was her. In regards to the Birthday, Miranda and Natalie asked Mrs Doubtfire to go with them so she wasn't invited directly by Stu. Considering Mrs Doubtfire made Miranda and the kids happy, Stu probably just went along with it.