However if they are only fasting until halachik midday, when they daven Mincha they do not say "elokeinu" [collections of the Mahri Vayil siman 41.] However according to what the Rama says (sief 2) one should say elokienu (he states that we say elokeinu in situations where if a cohen would go up to say birchat cohanim we wouldn't make him go down. So on this half fast we wouldn't make the cohen go down and therefore should say elokienu.)

2. (Magen Avraham comes to explain why we would daven at a different time on a fast day.) Throughout the year people want to eat after they daven mincha so they daven while it's still day (Ran.) And they for sure daven while the days still great (Suns shining strong) because it's prohibited from eating a half an hour before tzies (when three specific stars come out) like it says in siman 235. However, on a fast day (where there not eating during the day) they'd daven close to sunset.