How would hobbits do as wizards?

[The Barrow-Wight]

I think your last question points to the factor that would most affect how well they were received in Middle-earth. Throughout history, leaders were typically the tallest and strongest of the races with no notable (short) exceptions (at least none that I can remember). Thingol, Bullroarer and Elendil are three good examples. A Maiar in a diminutive body would like have less success in directing the efforts of the peoples of Middle-earth, and since the Istari were sent to deal mainly with the kingdoms of Men, the choice of Hobbit would have made no sense.

As for the idea of a Hobbit choosing a career in wizardry, I don't think it would be an option for them. They were remarkably 'unmagical' and thus, with the subtle magic of Middle-earth, they would have little possibility of wielding magical powers.