Is praying shacharis after chatzos (midday) permitted, and do you fulfill your obligation? If so, until when? Please give sources, preferably quoting the Mishna Berurah, Alter Rebbe of Lubavitch, or Rama.
See orach chayim siman 89, first siff in Ramma that once it hits chatzos its assur to daven. Mishna berura brings different opinions that if it was by accident one has a half hour after chatzos to daven, but one should assume like Ramma that there is no half hour make up time and therefore wait the half hour till mincha time, daven mincha, then pray a tashlumin for shachris. This applies only to one who missed zman tfila by accident. If one skipped a tephila on purpose there is no way to fix that.
Lechatchila person should not do it, if he already did in first half-hour after chatzos it is ok bedieved, but if already chatzos passee an one did not pray, he should do tashlumin after praying mincha.