Absolutely, a lot of people seem to misunderstand that the EAD is exactly what it is: an Employment Authorization Document. It has nothing to do with travel and you do not need it (nor are you permitted to use it) to reenter the United States as a parolee.
No, again the purpose of the EAD is to enable you to work in the US. It is not related to travelling. Applying for a new EAD is an optional benefit of your Adjustment of Status process, and leaving the country while it is pending has nothing to do with abandonment of your I-485 application. The only thing you need in order to NOT be considered as having abandoned your application is an approved Advanced Parole document.