I'm trying to connect a DAC/amp with a micro USB connection to my Android smartphone. Is it possible to get the DAC/amp to work without the larger "adapter" type OTG cable, and use something like this instead? (A USB micro male to USB micro male cable)
The red cable pictured is a USB-OTG cable too. It's no different to having the adaptor-style cable with a normal USB-A to micro-USB cable attached. So if your amp will work with one, it will work with the other too.
Just be extra careful when using a symmetric-looking USB-OTG cable. USB-OTG works by having the pins wired differently in the "host" and "peripheral" ends of the cable, so you need to be careful to plug the "host" end into the phone. Different cables have different ways of identifying which is which, so be careful to check the instructions that come with the cable.
Of course, your amp might not work for other reasons. If it lacks an external power supply, then it needs the host device to supply enough power for it. Most smartphones can't supply a lot of power via USB-OTG, so in that case you'd need to use it with a powered USB hub.
The other reason it might not work is if it uses the USB audio profile. Most Android devices lack driver support for USB audio; there are other questions about that on the site.