There’s second hand information about one of the sounds here (emphasis in all quotes added by me):
… the [Star Trek] phaser was derived from the hovering sound of the Martian war machines made for the 1953 version of Paramount’s War of the Worlds. The original was made with tape feedback of an electric guitar and a harp.
The Wikipedia article on the movie cites a print book (hard to verify) as the source for an explanation of another sound:
The distinctive sound effect of the weapon was created by an orchestra performing a written score, mainly with violins and cellos.
Later in the same article:
The machines also fire a pulsing green ray ... Its sound effect (created by striking a high tension cable with a hammer) …
Still later:
The sound effects of the war machines' heat rays firing were created by mixing the sound of three electric guitars being recorded backwards. The Martian's scream in the farmhouse ruins was created by mixing the sound of a microphone scraping along dry ice being combined with a woman's recorded scream and then reversed.
Looks like if you can get your hands on the book cited in the Wikipedia article, you’d have a solid source for all of your questions.
Here are the actual book citations:
- Warren, Bill. Keep Watching The Skies Vol I: 1950–1957. Jefferson, North Carolina: McFarland & Company, 1982. ISBN 0-89950-032-3.
- Rubin, Steve. "The War of the Worlds". Cinefantastique magazine, Volume 5, No. 4, 1977. A comprehensive "making of" retrospective and review of the film.