What's cooking?

You may have been thinking of 'battered' in its other (non-culinary) sense of 'beaten' ~ as in 'the ship was battered by the storm'. But in the context of food, 'battering' means 'coating a piece of food (usually meat or fish, but also doughnuts) in a thin layer of fatty and/or floury paste, and then frying it' ~ as in the classic Fish & Chips, of course.
If you were thinking of 'eggs that had been beaten (or even whipped) and mixed-up out of their original shape', you meant 'scrambled' or possibly an omelette.
In theory one might try to batter an egg, but this sounds rather a strange thing to do and probably wouldn't be worth the effort in terms of producing something tasty. This may well be why it's never offered!
The other three ways of serving eggs at breakfast are very normal, along with poached and as an omelette.

Dough is the 1st stage of baking bread, then the bread itself is cooked again to make toast.

In Answer 2, the banana is the odd one out since it is not a 'stone fruit' and does not readily grow in Britain.
Answer 1 are all herbs, Answer 3 are 'table birds' in various sizes, and Answer 4 are all from the spice rack.

Only Answer 2 seems a plausible sequence.
The overall sequence would probably be: Thaw ( = remove from freezer and allow to come to room temperature) ; wash / rinse ; chop/slice (and/or, perhaps, 'peel') ; heat (and/or 'boil') ; stir (and possibly 'cover and simmer', i.e. once it's boiled or boiling, you can leave it to continue cooking at that temperature) ; season ( = add flavourings, perhaps such as in Question 3, unless this was done earlier) ; drain ( = pour off the water once cooking has finished) ; slice ( = chop up the whole thing into portions before serving, as with a pie or a large joint of meat) ; decorate ( = garnish) ; serve. There are of course many other more specialised terms (grate, sprinkle, add, reduce etc.) but this sequence of a dozen or so should get you well on your way!

Answer 4 is probably the most likely, and practically comprehensive, selection (offering the standard 'cruet' plus tartare to accompany fish; mayonnaise for salads; and horseradish to accompany beef, or possibly smoked fish).
You would find vinegar fairly freely available in places that specialised in fried and/or grilled food (eg burger and chips, and similar), along ~ probably ~ with 'brown sauce'.
Worcester sauce is unlikely to be left on tables, but may well be offered at the bar or serving point to spice up a glass of tomato juice.

A cheese crumble would be rather unusual, though it would be possible (in theory) to dust the top of such a dish with grated cheese; but a 'meat (or fish) tart' would be most unlikely indeed.
You may well come across 'game pie' (i.e. containing the meat of birds or animals that have been hunted in open country), pork pie (with or without hard-boiled eggs inside it: 'gala pie') and even 'fish pie' (so-called; but the topping is much more usually done with mashed or sliced potato, like a 'cottage pie' ~ with minced meat, and perhaps some vegetables included ~ or 'shepherd's pie', which unsurprisingly would be made with sheep-meat [mutton or lamb]). Anything approaching a 'cheese pie' would more likely be known as a savoury flan or quiche.
All in all, a savoury tart (as such) would be possible but very, very unlikely ~ so Answer 4 is the one with the falsehood in it.

All (or certainly, most) of Answer 3 is true, but the crop which is milled into the porridge ingredient (Answer 1) is oat rather than barley.

Britain has an increasing number of sushi bars (Answer 4) but is only really 'hosting' these rather than adopting sushi within the national cuisine. Curry (often partnered with lager beer, from northern Europe) came to Britain via India in the days of the Empire, and now, with integration of many Commonwealth emigrants from the subcontinent, you are rarely much further away from a curry house than you are from a fish-&-chip shop ('chippy').
Tea (also from India, and/or China) and the sandwich (named after the Earl of Sandwich, who asked his servants to devise a meal that could be eaten straight off the plate with one's bare hands, so as not to interfere with a session at the gambling table) are both very British by long adoption. There is also the Cream Tea ( = tea + fresh scones with clotted cream or butter, and fruit jam).
Apple Pie meanwhile is sometimes quoted as essentially American, but we may find that questionable. The Dutch (in Holland/the Netherlands), among many others, were surely baking apple pies some while before European settlers established orchards in New England; meanwhile we doubt whether the aboriginals of North America (once, wrongly, known as 'Indians') were familiar with apple pie.

The items in Answer 2 are all reasonably natural and may well have been locally sourced (hence, fewer 'food miles').
In Answer 1, all the items for this apparent upcoming children's party consist of 'processed food' in one form or another: nothing fresh at all.
Answer 3 includes items from far away (bananas; food-miles again!), refined sugar, squash which almost certainly contains added ingredients such as flavourings, preservatives and 'acidity regulators', and two processed meat items. Kippers are smoked herrings, which according to some sources may slightly increase the risk of cancer (although many people find them delicious).
Answer 4 contains several further processed items; the 'economy chicken' was probably reared in a battery unit and nothing else is fresh. One might hope the people would at least be buying fresh vegetables ... !

Rice pudding is a milk-based dessert, and custard is a milk-based sauce, so one would be unlikely to try them together. Custard would more likely be served with another kind of pudding such as those discussed in Q.6 A.3 above.
All these other pairings are either long- or firmly established, or both; and if you haven't tried them, maybe you should! (Subject to availability, and your own dietary restrictions, of course.)