Have a look at the back of the connector box (Known as a Block terminal or BT) and check it dosn't say rf2 (RF3 is ok)
Typicaly this should be a BT78 (occasionaly bt80) you can open it to have a look inside (this bit of info is for anyone else reading this who's bt may be screwed to something rather than lying loose as this one is)
An RF2 can be identified by 2 small black cylenders inside about 4-5mm in diameter and about 6mm high. If you do have an RF2 then you need to call your ISP to get an engineer out to change this for an RF3.
Ther'e not common items but an RF would have been fitted if the normal telephony was being affected by some radio interference during a normal phone conversation. The RF2 was designed before Broadband was around and filters out the dsl signal completely unless it's corroded or your attenuation figures are so low because your close to the exchange the signal still forces enough through for your modem/router to still sync. Relacing it for an RF3 which has one small square black block with copper windings will do the same job but allows the dsl signal through with no hinderance.
Yes all this wiring belongs to bt so if you fiddle with it and you need an engineer to come out to put it right it will be chargeable
The black cable running into the black looks like a dropwire10 so this should be ok. the grey cable would be an internal grade of wiring and again usualy shouldn't be a problem as long as this runs straight into the nte. The only other problem which can arise is if you have star wiring, that is extention wiring connected before the nte as this can cause an inbalance which int turn can case a problem for DSL and also defeats the point of being able to remove the faceplate on an nte to test for normal telephony or dsl problems as some internal extention wiring is still connected.
If the BT has redcare on it or reads bt92 on the back, avoid it as this is for an alarm system and the phone monitoring needs to be deactivated before it's disconnected. Other wise put the kettle on for the boy's in blue as they'll round shortly