1) What are the parameters of ho'ochel b'shuk domeh lekalev (one who eats in the marketplace is like a dog) as we see in Gittin 61a that Rav Kahana ate in the open. Does this halacha apply to candy and gum also?
2) Also on 59a when it talks about Dovid having torah and gdula bmokom echad it answers that Dovid had Ira ha'yeiri. Why didn't the gemara say Mefiboshes or Shimee ben Gera or even Shmuel who are more commonly known?
1) Tosfos quotes Rabeinu Chananel that this rule applies only to someone who takes food that belongs to another person (in a way that he does not become Pasul for Eidus because of stealing) and eats it.
Rabeinu Tam says that this rule only applies to someone who eats a fixed meal in the market.
Rashi learns that we are referring to a person who has no embarrassement and would therefore potentially be capable of giving false testimony.
I think that according to all these explanations gum and candies would not be included.
2) Tosfos in Sanhedrin 36a (DH v'Ha) asks why the Gemara does not ask from Mefiboshes. Tosfos does not answer. The Maharam Shif in Gitin says that the Gemara could have asked from Mefiboshes or Achitofal, but chose one of a number of potential options. However, the Gemara could not have asked from Shmuel, because Shmuel was not alive when David's Gedulah began, that is when Shaul passed on.