I'll handle these questions in two parts: first, the connection between the words blessed and elect (including, as you request, a definition of blessed), and second, if the qualities listed are somehow indications of whether someone is elect or not.
Connection between blessed and elect
It's very clear to Calvinists, and, I suspect, to at least most Christians, that Jesus here is referring to people who are saved, and to Calvinists, that means that they are elect. There are a couple reasons for this:
- The definition of "blessed"
- The promises given, like "the kingdom of heaven" and "they shall see God"
Regarding the definition of "blessed," the Reformed Study Bible writes:
This means more than the emotional state represented by the word "happy." It includes spiritual well-being, having the approval of God, and thus a happier destiny (Ps. 1).
The "blessedness," then, is a single idea that the passage describes in several complementary ways. Similarly, Calvin identifies the point of this passage to be "true happiness": not the happiness as understood by the world, but a happiness that looks to the promise. Matthew Poole writes regarding verse 3:
true happiness lieth not in worldly possessions, but in the favour of God, and a right to the kingdom of heaven, and that these men have. (source)
The promises, then, are the spiritual blessings that are inherent in being "blessed," and are indicative of salvation. For example, Poole writes regarding those who mourn "their own sins" in verse 4:
They shall be comforted, either in this life, with the consolations of the Spirit, or with their Master’s joy in the life that is to come
Regarding verse 3's "poor in spirit," Calvin understands these to be those who rely on God alone:
It deserves our attention, that he only who is reduced to nothing in himself, and relies on the mercy of God, is poor in spirit. (source)
John Gill writes that for these, "[eternal glory] is prepared for them, and given to them; they are born heirs of it, have a right unto it, are making meet for it, and shall enjoy it." (source)
Qualities indicative of election?
Calvinists take two similar but distinct approaches on the universal applicability of the qualities listed in this passage.
Qualities are responses to affliction
The first approach, taken by Calvin, is to understand these qualities as primarily responses to affliction. Thus, the poor in spirit and those that mourn are suffering under some trial, and the promise is that they will be blessed for it. By extension, then, these qualities will not necessarily be visible at all times, since the life of the Christian is not constant suffering. As Calvin explains:
But if, at any time, the Lord spares our weakness, and does not permit the ungodly to torment us as they would desire, yet, during the season of repose and leisure, it is proper for us to meditate on this doctrine, that we may be ready, whenever it shall be necessary, to enter the field, and may not engage in the contest till we have been well prepared. As the condition of the godly, during the whole course of this life, is very miserable, Christ properly calls them to the hope of the heavenly life. (source)
Thus, to Calvin, periods of respite from affliction are times when the opportunity to display some of the qualities listed here may be diminished. That doesn't call into question one's election, rather, it serves as a time of preparation for future trials.
Qualities are universally applicable
Other Calvinists more universally apply many of the qualities listed in the passage. For example, John Gill writes that those who mourn are mourning "for sin, for their own sins [...] the unbelief of their hearts [...] [and] the sins of the world" in addition to mourning "under afflictions." (source) Thus, the quality of being "mournful" more generally applies to all of the Christian life, not merely during trials.
Similarly, Charles Spurgeon understands the "poor in spirit" to be "not those who boast themselves of spiritual riches and personal goodness, but the lowly, the meek, the trembling, the humble." (source) Again, the qualities are not limited to trials.
It's worth mentioning that both groups of Calvinists recognize that Christians will continue to sin, and will not always display the qualities described in this and other passages. Failing to have these qualities would thus not necessarily be an indication of not being elect, but growth in them would be expected.
Calvinists view Matthew 5 as referring to the elect and the qualities that they will develop more and more as they grow spiritually. Some emphasize that these qualities will be visible primarily during periods of affliction, while others suggest that most of them apply generally to the entire Christian walk.