Electronic – Increasing the opamp gain is decreasing the DAC resolution

Simple use case: you want an output higher than the internal VREF (2.048V) but don't want to add to your parts count with an external reference.

The less obvious case is what the datasheet calls "multiplier mode". In this case you keep the digital signal fixed and feed an analogue signal into VREF. That gives you a programmable gain amplifier, and the 2*VREF mode lets you have a gain between 0 and 2. That could be useful for analogue control loops or sensor biasing.

Other answers have already made the main point: TL084 is not a good op-amp for your application. I'll answer one of your specific questions:

And how does the input offset voltage appear at the output (suppose that the input offset voltage s 3mV, the input signal is x(t) and the opamp output is y(t))?

The input offset appears as if it were a voltage source in front of one of the input pins (doesn't really matter which one).

This typically means it appears as an error in your input signal, and sees the same gain as the input signal.

a) 1000 times amplified along with the input signal. (i.e.; y(t) = 3V + 1000x(t))

Yes, though it might be more clear to say \$ y(t) = 100 \left( x(t) + V_{os}(t) \right)\$.

If you look you will be able to find an op-amp with V_os in the 10 uV range, which might be acceptable for your application.

You might also consider that 1000x is a very high gain to achieve with a single op-amp stage. You might want to consider doing this with two stages instead of one.