Is one allowed to eat if he needs to use the bathroom? Does it make a difference if he made a Bracha yet?
The needing of the bathroom is divided in categories by Halacha. In Shulchan Aruch Orach Chayim siman 92, you can find all those halachot.
If one needs to bathroom and cannot contain himself the time needed to walk a distance of one parsa (72 minuts), he is not allowed to continue the prayer , blessings or learn Tora. But if one feels a need before starting, he cannot start to pray.
If one is able to contain himself for this amount of time, it's right to continue the Tefila and to go in bathroom afterward, it's better than interrupt the prayer. (Poskim in name of Teshuvot Harashba).
Concerning eating, if one needs to use the bathroom, I did not find anything in Hilchot Derech Erets Baseuda (OC 168-171).
See Seder Hayom (סדר בדיקת עצמו קודם האכילה ונטילת ידיים"") that says to check oneself before the meal and which who reports the Gemara in Masechet Shabbat 41a,
One who has a call of nature yet eats is like an oven which is heated up on top of its ashes, and that is the beginning of perspiration odour.
Note regarding prayer. Further I learned in Shulchan Aruch OC 92 in Magen Avraham sk 1 that (see Bait Chadash on Tur in the same siman before the reading of Magen Avraham) accordind to the Rif (and Rambam) if he can refrain himself up to the end of prayer, and the time of tefila is finishing, he can pray lechatechilla. The MA stated as the Rif. So, for birkat hamazon this is obviously the same if the time is ending.