Uhh, no they should definitely keep them. If they change it so that Annasui's first appearance in Part 6 is as a man and not as a woman for example, that would be stupid, because it's hilarious.
I can hardly call Iggy's facial change a mistake, even. Character designs change all the time, usually in more subtle ways, but changing Iggy's face to be more human-like so he can show emotion during the fight is a good thing.
The reason Iggy never did anything is because he's a dog and doesn't give a shit about some evil vampire in a fight he was dragged into. Only when he was directly threatened did he start caring, because that's how Iggy is.
In future parts the cast gets more screentime and does more, for the most part. In Part 3 Jotaro probably beats 75% of the enemies, and Avdol, Joseph and Iggy hardly get any solo fights, which sucks. It's an improvement from Part 2 where Joseph defeats literally every single major enemy, however.
I've heard about the famed gender change of one of the characters in Part 6. I'm still unsure of the circumstances (I've heard it was editor pressure to avoid a homosexual relationship involving the main character which is lame as hell if true) but it that isn't addressed in the anime adaptation of Part 6 (as in the character is introduced and stays male) that isn't funny, that's STUPID. Anime adaptations shouldn't be there just to regurgitate the manga/novel but to capture the essence of what makes those things popular/good and present it in a manner most befitting the realm of animation. That means should be fixing obvious plotholes or art errors that don't have cascading negative effects on other aspects of the adaptation.
Regarding character design, I'm well aware that there is some degree of fluidity to them, especially after long periods of time. I don't expect the Jotaro of Part III to look exactly the same as the Jotaro of later Parts. I don't expect Araki to be able or interested in drawing Johnathan Joestar the same way in 2015 as he did1986. However, that start of the Egypt arc of Part III and the Petshop fight can't be more than like 6-8 months apart right? The change in Iggy's design is DRASTIC.
I understand why it was made - as you said, it is easier to humanize and characterize Iggy with a humanoid face and unlike say Toothless in HTTYD, manga is not a medium that would allow Araki the same tools to characterize Iggy 20+ years ago. What either happened is that Araki didn't intend to really humanize Iggy when he was introduced but changed his mind later in the arc (a bit of making things up as they go along that a lot of manga authors of long running series run into) and was forced into the character design or knew where he was going to take Iggy from the start of the arc but didn't think or pay much attention to how he was drawing Iggy to start with. Either way, as an adaptation of the material some 20+ years later, EVERYBODY already knows where Iggy's character is going to go. "Man-Face" Iggy is the culmination of his character and as the anime team has the hindsight of decades and the ability to retroactively "fix" this issue, the fact that they didn't screams the fact that they were more interested in following the manga to the letter, warts and all, than doing what makes the most sense. There are aspects in which I really appreciated that they stuck close to the source material but anyone doing an adaptation should know when to stay close to the source material and when they can deviate and make improvements. Blind adherence is not something to be lauded.
As for character "usage", I agree that its a shame that nobody aside from Joesph got to do much in Part II. However, considering how much longer Part III is than Part II, that we go from a relevant cast of 3 to 6, and in Part III cast members are effectively written out for long periods of time, I'm becoming less convinced that Part III used its cast better than Part II. Or at least the difference is smaller than I would like.