Language Challenge

'Gay storm bishop' is a journalistic way of identifying a bishop who has been involved in a 'storm' (i.e. a major public row) about homosexuality ~ without clarifying which side of the argument he is on (i.e. whether, as a Christian, he is broadly sympathetic and tolerant towards such people; or whether, perhaps, he has been outspoken in saying he feels they have no place in the church).
'Slams' does not literally mean that he has been slamming doors; it is another short, impactful word (like 'hits', earlier) signifying that he has made a strong negative criticism ... not of the Gays, but of the new doors, which he may personally feel are too bland, or ugly, or in some other way out of keeping with a (presumably) traditional cathedral building.
(Perhaps he feels the doors are too flamboyant, or 'camp', in style; in which case, we might in turn sense that he does not have much sympathy with anything colourful or stylish, maybe including the clothing and behaviour of gay people.) Not a very likely story; but probably the least unlikely of the versions offered. We would stress that at time of publication, this is/was an entirely fictitious example!

There are plenty of 'bridge' puns here (journalists love such wordplay); but the main topic is the building of an overpass, such as to carry a major road across a river or railway line. One company's 'bid' to do this work has been defeated by another company that offered a better price, at a late stage in the open competition

It obviously helps if you happen to know that Reading (pronounced with its front syllable like 'red', the colour) is the county town of Berkshire, about 50 miles west of London.
From there we can unravel the fact that this 'gang' specialises in robbing jewellers' (the classic shops, since their goods are usually particularly small and portable and valuable); and that the police are 'holding' them temporarily in custody, presumably in the cells at a local police station.
This example shows how several quite short words (which can then be printed large, to catch the reader's eye) can encapsulate a fairly complex story

'Strikes in connection with the buses have had a negative impact on a band'
'Hit' is a short, sharp word meaning that one thing that's happened has a direct, and presumably bad, effect on something else.
Meanwhile it's not the buses that are on strike, of course, but the people who would otherwise have been driving them. This too is a kind of 'shorthand'; but 'bus strike' would be fairly familiar and understandable to the British travelling public

The woman involved is clearly singular and 'local' (to wherever the paper is published; but the jail she is in may happen to be in some other part of the country)

The van is a vehicle belonging to the organisation that looks after the royal parks. It has been involved in some way in an unusual incident in or around a lake ~ such as that it happened to be nearby, and was used to shelter &/or transport a visitor who had just been rescued from drowning when a small boat capsized on the water.
No member of the Royal Family may have been anywhere near the scene, nor ever have had anything to do with the van; but even an indirect 'Royal' reference may attract the interest of certain kinds of newspaper reader!

'United' is often used as 'shorthand' for the full name of a football club (such as in Manchester, Leeds etc.). 'Shares' is a noun here (as in 'stocks & shares'), so 'United shares ... ' signals a story about the value of shares in the club which have been 'plunging', and people are 'shocked'.
Again, the alternatives are (to a greater or lesser extent) plausible, but less likely

'Flies' is the plural subject of the verb 'close', so the story is shaping that the closure of the restaurant has been caused by the flies (these may be live insects buzzing around, or dead ones found in the kitchen or even in the food).
The interpretation in Answer 2 has got off to a mistaken start by misreading 'flies' as 'files' ~ easily done at a moment's glance!

'Fine' is usually an everyday adjective, as in 'fine weather', 'the fine arts' or 'fine wine'; but here, in a paper, it could at least equally refer to the fact that a public servant (who should set a law-abiding example) has been penalised for breaking some rule. 'Seat' probably refers to his 'place' at council meetings; the fact that SEAT happens to be a car brand (and indistinguishable within a headline printed all in uppercase letters) is probably a distraction.
Answers 1,2 and 4 seem less likely or more trivial interpretations; but they remain possible

'Yard' is short for 'Scotland Yard' (= police HQ), though often a short headline may refer to them as 'cops'. Someone is a 'suspect' because the police believe he may be guilty of trying to set light to stations on more than one occasion; we are not told (yet) whether these were police stations or railway stations, or indeed some other kind; but clearly, the detectives are seriously interested in putting a stop to dangerous disruptions to public institutions