Quisiera is more formal while me gustaría is used in normal daily conversations.
I was told to use Quisiera when speaking with someone with more achievements than me, with someone on higher position or where You have to be polite (like my boss in work or stewardessa in a plane).
Me gustaría is used in normal conversations with friends, in restaurants or for example in the shop when You ask the vendor for different but similiar item.
The acronym WEIRDO stands for Wishes, Emotions, Impersonal Expressions, Recommendations, Doubt/Denial, and Ojalá, which are all situations in which you're likely to use the subjunctive.
You could make the argument that ojalá is a subset of "wishing" as it's from the Iberian Arabic version of inshallah, but WEIRDO is probably catchier and easier to remember than WEIRD.
Because the subject is yo. It is not always quedarse, it depends on the subject. Did you see before that there were words that were reflective like: yo me pongo, tu te pones, nosotros nos ponemos? If you use the infinitive the word is still reflexive and you should put it connected to the end of the infinitive. To explain it better, subject vs reflecive form: yo quiero quedarme tu quieres quedarte él/ella quiere quedarse nosotros queremos quedarnos éllos/ellas quieren quedarse Hope that clears it up :)
In the simplest (though somewhat misleading) terms, use "me gustaría " when describing something you "would like" to do (me gustaría ir a España) and use "quisiera " when you "would like" to request something (quisiera una copa de vino tinto de España).
I say it's somewhat misleading because in many instances they can be used interchangeably and the usage varies regionally. For example, in Spain, both are usually viewed as overly polite and would sound strange in a typical context (i.e., when you aren't speaking to the king and queen).