Will doxycycline affect sprentic birth control pills?

Yes. Birth control pills are all at risk of being affected by antibiotics like doxycycline. You should use a backup form of contraceptive like condoms or abstinence for the rest of this cycle.
No interference. It's an outdated urban myth that antibiotics interfere with effectiveness of birth control pills. They do not, except occasionally with rifampin, an infrequently used antibiotic for certain infections like tuberculosis. Doxycycline certainly has no effect. See https://www.drugs.com/article/antibiotics-and-birth-control.html.

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Will I get a yeast infection taking doxycycline, prednisone and birth control pills all at the same time?

Yeast infection. Antibiotics can predispose one to a yeast infection by killing off the "good bacteria" that would normally keep yeast at bay. Steroids can add to the problem by decreasing some the body's natural immune/inflammatory response to the yeast allowing it to proliferate. Read more...