Differential Amplifier Biasing Problems
There are a number of things wrong here, but the most fundamental is that you have positive feedback.  That's why the output is slamming to one rail.

You didn't say, but I'll take the base of Q1 to be the input to this circuit, and the output the emitter of Q3.  The signal is inverted from B of Q1 to C of Q1.  Q3 provides more drive current, but doesn't invert.  The output is therefore inverted from the input.  This means the base of Q1 is the negative input, which makes the base of Q2 the positive input.

You then feed back part of the output to the positive input.  DC positive feedback causes latching behavior, which is exactly what you see.

Added in response to comment

Yes, the positive and negative inputs to the differential front end of this amplifier are the bases of Q1 and Q2.  Each effects the final single-ended output with opposite polarity.

There can be multiple inversions between the diff amp input and the final output.  That only effects which one of the two inputs is considered the positive and which the negative.  If done right, they are essentially symmetrical, so you simply flip their usages to accommodate a net inversion.  Note that changing which collector the single-ended signal is take from also flips the polarity.