I can’t speak for musical recording, but I can for video production (private and light commercial)
If we start with the basics, the mic type/design will make quite an impact on your sound. While you don’t have to spend a lot of money, the audio differences are quite noticeable and should be selected based on your taste, vocal style and recording environment. (for example, I had 4-5 mics with me on a shoot) Also, you will want to take into account your “sonic preferences”. Within a given selection of 5 mics, 5 users might rank them completely differently. There are many good quality mics for generic portable devices like Android phones and pads that won’t break the bank, but you will need to dig a little and check into the actual Android hardware. It you do go for a phone based analog mic (no experience with USB into phones), be sure you get one designed for a phone audio jack…typically TRRS ( Understanding TRRS and Audio Jacks - Cable Chick Blog
The recording environment will play a big role as well. While there are post capture ways to clean up the audio, depending on your circumstances, getting it right @ the source can make a big difference.
As to stand-alone recorders (field recorders), there are some great options out there that won’t break the bank and I feel are the best choice. An oldie but goodie for about $100 is the Zoom H1n Handy Recorder. It is an older unit but a staple for many INDY and even pro producers. The UI is a bit clunky but it is small, has a bunch of great features, killer pre-amps and allows for external mics and headsets. It even has a way to mount it to a mic stand. Some might squawk about it’s age, digital resolution or lacking an XLR jack, but I’d bet a bottle of Sassicaia that they could not tell the difference 6 out or 10 times.
With regards to synching, these files are typically just standard audio files (wav, ogg, etc) but my experience with synching audio is related to video streams so I’m not sure if there are any “special” issues to consider. However, just like synching an audio source track with other analog or digital tracks in my DAW, I can’t imagine there are any real differences or issues that can’t be overcome in the mix.
Like all things creative…this is just one opinion amongst many and hope it helps. Good luck.