Factors in play for Yahoo this week could decide whether the pioneering Web company can remain independent or must surrender to an unsolicited takeover by Microsoft.

Yahoo is racing to forge a credible alternative that lets it stay independent or at least pushes Microsoft to raise its $31 a share cash-and-stock bid, now valued at $42.8 billion.

“Yahoo is willing to try things at the 11th hour that it never felt the urgency to try,” according to the Sanford C. Bernstein analyst Jeffrey Lindsay. “Shareholders win, either way.”

“They are coming up with some of their best stuff now,” he added. “We just wish they had done these things last year.”

Yahoo reports first-quarter results on Tuesday, and has perhaps a last chance to demonstrate a degree of the financial strength and progress it has made in stabilizing the company’s Internet media and advertising business after two years of decline.

By midweek, Yahoo is planning to complete a test with Google on whether Google should run a piece of its Web search ad sales.

Saturday is the deadline Microsoft has set for Yahoo to accept the deal or face a drawn-out proxy battle to unseat the Yahoo board. Two weeks ago, Microsoft threatened to lower its offer if Yahoo did not conclude friendly merger talks by the end of this week.

The chief technology officer for Yahoo will deliver a speech Thursday at the Web 2.0 Expo industry show and is expected to spell out a strategy to open up Yahoo services like e-mail, news, sports and advertising to make them more relevant across the Web, not just for users drawn inside its own sites.

That same day, Microsoft reports its own quarterly results. It is expected to show strong underlying fundamentals across its range of businesses.

But despite the time pressures, and indirect threats by Microsoft…