is it normal to see cobwebs/insects in there? Could this bugs cause internet issues if they are crawling around on the wires?
This was the rather nasty interior of my BT66 when it was opened. It hadn't been touched since the house was built appx 1993. The OR engineer wasn't at all concerned about the amount of dirt and cobwebs which he said wasn't too unusual. In my case it was oxidisation of the joints that was causing SNR fluctuations on the upstream when the phone rang. The copper wire literally fell apart when he poked with his screw-driver.
So to answer your question about insects, then yes a spider crawling about over any of those particular joints which are badly corroded could easily have disturbed them and cause fluctuations in the SNR. However, newer installs no longer have those metal IDC blocks, they use jelly crimps to make joints thus eliminating what is/was a common fault point due to oxidisation.
As tubaman says, it depends on the joints. As you have jelly crimps you should be ok.