Hellenists

Who are the Hellenists and the Hebrews?
As referred to in this passage, "the Hellenists" are the Christian converts among the Jews who had returned to Judea after having lived abroad in the Greek world. Despite being Jewish, the Hellenists had adopted Greek cultural elements and spoke Greek. "The Hebrews" in this verse are the Christian converts among the Jews who were born and raised in Israel.

Why were the widows among the Hellenists "neglected in the daily distribution" (Acts 6:1)?
The church was going through some growing pains of trying to keep up with everyone in the midst of rapid growth (see Acts 6), and had neglected, most probably without intent, the widows among the Hellenists.

Why wasn't it desirable for the twelve Apostles to "leave the word of God and serve tables" (Acts 6:2)?
With only twelve apostles to tell the tens of thousands of new and multiplying Christians about Jesus, the most pressing need was "the ministry of the word." The apostles weren't saying that manual labor was somehow below them, but that telling people about Jesus couldn't be held up while they perform roles that others can perform (In fact, in this context, to "serve tables" didn't even mean to serve food on tables, but to 'tend to 'business,' which were conducted on tables, of church management). Each Christian is given gift(s) by God and we should focus on using them to glorify Him: "For as we have many members in one body, but all the members do not have the same function, so we, being many, are one body in Christ, and individually members of one another. Having then gifts differing according to the grace that is given to us, let us use them: if prophecy, let us prophesy in proportion to our faith; or ministry, let us use it in our ministering; he who teaches, in teaching; he who exhorts, in exhortation; he who gives, with liberality; he who leads, with diligence; he who shows mercy, with cheerfulness." (Romans 12:4-8)

Which of these roles deserves particular esteem?
All of them: "For as the body is one and has many members, but all the members of that one body, being many, are one body, so also is Christ. For by one Spirit we were all baptized into one body - whether Jews or Greeks, whether slaves or free - and have all been made to drink into one Spirit. For in fact the body is not one member but many. If the foot should say, “Because I am not a hand, I am not of the body,” is it therefore not of the body? And if the ear should say, “Because I am not an eye, I am not of the body,” is it therefore not of the body? If the whole body were an eye, where would be the hearing? If the whole were hearing, where would be the smelling? But now God has set the members, each one of them, in the body just as He pleased. And if they were all one member, where would the body be? But now indeed there are many members, yet one body. And the eye cannot say to the hand, “I have no need of you”; nor again the head to the feet, “I have no need of you.” No, much rather, those members of the body which seem to be weaker are necessary. And those members of the body which we think to be less honorable, on these we bestow greater honor; and our unpresentable parts have greater modesty, but our presentable parts have no need. But God composed the body, having given greater honor to that part which lacks it, that there should be no schism in the body, but that the members should have the same care for one another. And if one member suffers, all the members suffer with it; or if one member is honored, all the members rejoice with it." (1 Corinthians 12:12-26)

Should teachers only teach?
Not necessarily. Teaching should remain the focus of those truly called by God to teach, but they aren't precluded from "serving tables" in church or even working to provide for themselves. After all, Jesus washed His disciples' feet, and apostle Paul continued to work as a tent maker during his ministry (see Aquila and Priscilla).