People under the age of 18 years can't enter into a legal contract, so it's unsurprising that 16-year-olds experience difficulties. Package deals are likely to be very difficult, if not impossible, to book. However booking each element individually should present no problems.
Ryanair carry unaccompanied 16 year-olds:
http://www.ryanair.com/en/terms-and-conditions#regulations-childreninfantsyoung
(They serve Berlin Schönefeld from East Midlands & Stansted).
easyJet is another possibility:
http://easyjet.custhelp.com/app/answers/detail/a_id/4300
(easyJet serves Berlin Schönefeld from Edinburgh, Liverpoool, Luton, Gatwick, Manchester and, from June 17, Southend).
Jet2 is yet another option:
http://www.jet2.com/new/answers/travelling-with-children-and-animals
(Jet2 serves Berlin Schönefeld from Leeds/Bradford).
There are also no problems with BA:
http://www.britishairways.com/travel/childinfo/public/en_gb
(BA serves Berlin Tegel from Heathrow).
Lufthansa is OK as well:
http://www.lufthansa.com/online/portal/lh/uk/info_and_services/travel_preparation?nodeid=1770716&l=en
(Lufthansa fly to Berlin Tegel from Heathrow and, using Eurowings, from Birmingham).
There should also be no problems with Air Berlin:
http://www.airberlin.com/en-GB/site/faq.php?et_cid=128&et_lid=183&et_sub=eng_header_faq
(Air Berlin serve Berlin Tegel from Aberdeen, Edinburgh, Glasgow, London City, Heathrow, Stansted, Manchester & Newcastle).
There are 89 hostels, 31 budget hotels and 10 B&Bs here
http://www.hostelworld.com/
all of which should accept young travellers.
The links here can provide information about bus and rail services, from both airports, into the city:
http://www.berlin-airport.de/EN/ReisendeUndBesucher/AnUndAbreise/AnfahrtBusUndBahn/SXF/OeffentlicheSXF.html
Chris