Need Advice installing new ceiling fan

Yes you can splice into the wire. But, most likely, there won't be long enough to get the 2 sections in the new (fan rated) ceiling box and have 2" of wire sticking out.

So you will either need to install an addition box to use as a junction box or to run a new cable from either the switch or the receptacle.

And if you run a new cable from the switch then you can run 3 wire cable so that you have independent control of the fan and light.

But you need to verify where power comes in and how the switch is wired.

You also need to look at the receptacle and see what connection are there.

Are both halves of the duplex receptacle switched.

There are many different ways that could have been wired.