Microphone only works in default browser and no other programs

Nircmd's setdefaultsounddevice command should help you:

nircmd setdefaultsounddevice [Device Name] {Role}

Set the default sound device on Windows 7/Vista/2008. The [Device Name] is the name of the device, as it appears in the sound devices list of Windows, for example: Speakers, Line In, Microphone, and so on...

The {Role} parameter is optional and may countain one of the following values: 0 for Console (the default value), 1 for Multimedia, and 2 for Communications.

Examples:

setdefaultsounddevice "Line In"
setdefaultsounddevice "Microphone" 2

You can easily create a shortcut for this manually or using either cmdshortcut or cmdshortcutkey.

This AutoHotkey forum thread also contains scripts that you might find useful.

Finally, Set Sound Device is a compiled AutoIt script that helps you do the same thing:

SSD enables you to change the default Sound Device for Win7 (works maybe for Vista too, but this is untested) via the commandline. To call SSD from the commandline, the syntax is ‘SSD.exe #’ (where # is the number of the Sound Device to select from the list,’Select Default’ dropdown box must be enabled to make device selectable).

In Zoom, go to share screen -> Advanced -> Desktop audio, and click on the share button. This will share any audio you can hear over your headphones, except the voices of other participants in the zoom call. read complete tutorial with images, or see the Zoom help page

If you need to show images, too, you'll need to share your desktop, or in the advanced tab select portion of screen. Then make sure the share audio checkbox is ticked before clicking on the share buttton.

In the end, make sure your mic is on in Zoom, so others can hear your voice.

EDIT

1. Note that when sharing sound, default is "mono", but you can choose "Stereo" from the drop-down menu next to the sharing option.

2. Zoom will apply various filters to any audio. That's why the OBS / audio-cable won't work: Zoom will filter out any music or anything it classifies as "noise" from the mic. However, there's a way to transmit exactly the input audio: In Settings -> Audio, enable the Show in-meeting option to enable "Original Sound". Then, during the meeting, you'll find at the top left a button to switch to Original Sound. Doing this, you might be able to use the virtual audio-cable, if for any reason you prefer to do it that way.