The Democrats' mid-term conference, which opens in Philadelphia tomorrow in the pursuit of party unity, already has produced a few preliminary cosmic mini-struggles among the men who would be the nation's next president.

First was the skirmish over the order of the speakers. Then there was the tug-of-war over the TelePrompTer.

The seven unofficial 1984 presidential candidates already have made more than 475 unofficial campaign stops in all 50 states. The mid-term conference assures them, for a change, of a guaranteed audience--987 delegates, or participants as they're called this year, about 1,000 journalists and an estimated 3,000 or 4,000 politically active and interested onlookers.

Early last week, things seemed to be going smoothly enough for Democratic National Committee Chairman Charles Manatt, who has changed the name of the affair from mini-convention to conference because the word convention has a connotation of competition and conflict.

He had invited all potential candidates for president to address the convention, in an order no one quarreled with, based loosely on the protocol of politics. They would all speak at the first plenary session tomorrow afternoon, right after the traditional opening remarks by the party's congressional leaders and others.

Former vice president Walter F. Mondale would begin, followed by Sen. Edward M. Kennedy, who had commanded many presidential delegates at the last party convention. Then would come the rest: Sens. Ernest F. Hollings of South Carolina, Alan Cranston of California, Gary Hart of Colorado and John Glenn of Ohio.

Reubin Askew, although he will be politicking at the convention, declined the invitation to speak "because I am trying to keep my effort very low key."

But then Senate Minority Leader Robert C. Byrd told Manatt he had decided not to come at all, and designated his Senate whip, Cranston, to speak for him--which meant that Cranston would be speaking ahead of all the others. And it all fell apart after Manatt communicated this to Kennedy's assistant, Lawrence Horowitz.

The Kennedy aide objected and expressed surprise, as Democratic sources tell it. Cranston, among other things, is a staunch advocate of a freeze on nuclear weapons deployment, and Kennedy has made the nuclear freeze one of his major crusades--he even sent each delegate to the mini-convention a book about it.

But Kennedy's aide did not express concern that Cranston would steal Kennedy's nuclear thunder. Instead, as one Kennedy aide said, "we were concerned because we did not want to be in the middle of a long day of speeches."

Horowitz voiced concern about how late the Senate might stay in session tonight--which the Democratic officials took as a signal that Kennedy might boycott the conference altogether. In a flurry of conference calls, a compromise was reached. Kennedy would address the convention Sunday instead.

Manatt gave all other prospective candidates the option of switching to Sunday. They declined. Now Kennedy alone will have the opportunity to address the convention on Sunday--"just before the convention votes on the nuclear freeze resolution," a Kennedy aide noted, adding that Kennedy also has the Sunday night television and Monday newspapers all to himself.

What the Kennedy adviser did not know was that, even as he was speaking, Byrd was contacting party officials to say that he would be coming to address the mini-convention after all--on Sunday, before Kennedy.

The skirmish over the speaking schedule, meanwhile, led directly to the controversy over the TelePrompTer. Kennedy wanted one; the others did not.

Nor did they want to pay for one. Kennedy aides say the Democratic National Committee informed him that the device was too expensive.

Kennedy aides checked and found it would cost $1,100 for the entire convention, plus $32 an hour for two operators--Democrats will pay more for frozen shrimp at their cocktail parties. But no matter. If the party will not pay for it, Kennedy will--for Sunday use only.

To assure that there will be no discordant notes, the Democrats have appointed a music coordinator for the event, with each candidate allowed to choose a theme song to be played upon his entry.

Cranston chose "The Theme From Chariots of Fire"; Mondale the "Minnesota Rouser Fight Song"; Glenn the "Ohio State Fight Song"; Hollings "The Charleston"; Hart "On the Road Again," and Kennedy "Happy Days Are Here Again."

For the men who would be president, the mid-term conference can seem like one big singles bar, but it means different things to different ones.

For Kennedy, the frontrunner in the current polls, it is the chance to promote party unity in the hopes that he will be seen at last as the rightful heir to the party leadership.

"He sees the mid-term coming at a very fortuitous time," says one of Kennedy's top advisers. "After a very difficult period, the Democratic Party is starting to jell. It is coming together. And if there are any wounds, any grudges from the 1980 primaries, he does not share them."

Kennedy has traveled to 17 states in the past two years and, through a political action committee, is contributing to the 1982 campaigns of liberal Democrats.

For Mondale, it is a chance to demonstrate that he has strength among the party officials and loyalists that transcends his modest showing in the polls. He too has a political action committee dedicated to helping other Democrats campaign in 1982 and, like Kennedy, building IOUs for 1984 in the process. He has traveled to 29 states and has helped the campaigns of 55 candidates so far this year.

For the others, it is a chance to demonstrate that they deserve to be considered as significant candidates for president.

To this end, Glenn and Cranston, along with Kennedy and Mondale, will have trailer headquarters installed in the Civic Center. They are intended not so much to coordinate convention floor strategy--there probably will be little need for that--but to impress delegates and the media.

Glenn, the recent subject of a cover story in Parade magazine, has visited 28 states in the past two years, including obligatory visits in Iowa and New Hampshire, the first caucus and primary states.

Cranston has been to New Hampshire nine times and Iowa seven times in traveling to 22 states this year. He has formed a presidential exploratory committee that has raised about $50,000 to finance his expenses while he is officially deciding whether or not to run.

Hollings has visited 17 states this year, including stops in the past month in Iowa and New Hampshire. He, too, has a political action committee to funnel funds to candidates whom he designates; it has raised about $150,000 so far, most of it as yet unspent.

Hart has neither a political action committee nor an official exploratory committee to underwrite his decision-making. A "Friends of Gary Hart" bank account containing about $15,000 in donated funds serves almost the same purpose, however, financing trips to 16 states this year.

He is hosting a workshop for all participants on "how Democrats can run--and win--on issues."

What Askew calls a "low key" exploratory campaign has--to the surprise of many party pros--taken him to all 50 states in the past two years, often just meeting people in small groups and leaving again without so much as a public speech.

His committee has raised $230,000 and spent $150,000 on his travel, expenses, and the salaries of five full-time and four part-time employes. He says he has been doing so well that he will have to be convinced not to formally announce his candidacy by the end of this year.