Help installing Nest Smoke/Co alarm to capped ceiling light wires
Based on information from Nest's website, the Nest Protect doesn't have wired interconnect, only wireless, so you shouldn't have to worry about the interconnect wire - unless your user manual says something differently?

I'm guessing that the white/braided wires with the white cap is your neutral line. Since there are three hot wires running through your junction, there would likely be three neutrals as well. This can be confirmed by:


Disabling power to your circuit (turn off breaker)
Disconnecting the wires under the white cap
Turn power back on and check to see if any lights don't work
If so, that's your neutral


As for joining into the black wire - I would try to remove the copper connector and then tie in your Nest's wire using a new wire nut. If the connector is crimped on too tightly, you'll have to clip the wires back just behind the connector, re-strip them, and then install the new wire nut.