How to record

The only white butterfly I'm personally happy to record if I've only observed it in flight is the male Orange-tip, anything else you've just got to accept is not of a sufficient level of certainty to record, it's very frustrating. You can of course record butterflies you get a good visual sighting of, if they only settle briefly, but don't manage to photograph so long as they are distinctive, like an Orange-tip or a Green-veined white for instance (if you get a good look at their underwing).

I think your post covers a few different but related points:

• You should aim to record things where you are confident of the ID. So in your first example if the white butterflies could have been Small or Green-veined White, then you can't really record them (you could record them to genus only, but the records aren't particularly useful). In the birds example, I wouldn't rely on Merlin, so if it is telling you a particular species is present then you see it and are happy with the ID, then record it.
• If you see 10 unidentified white butterflies then photograph and ID two Green-veined Whites, just record 2 Green-veined Whites don't guess that they might have all been that species.
• Photos give useful corroboratory evidence of sightings, however you can still record things without a photo. Whether they are accepted will then be down to the verifier, but for example if you have submitted quite a few correct bird records with photographs in the past, the verifier will be able to see that and is likely to accept records of common species as "considered correct"

It should be possible to enter just the genus (so Pieris if you only have large/small/green-veined to worry about) or family (so Pieridae if you also have female orange tips or wood whites in the mix).

In the Garden Butterfly Survey specifically, it is possible & acceptable to record white Butterfly IDs at genus level, i.e. Pieris. Butterfly Conservation have set up this Survey with this as an ID option.

As a verifier, I cannot accept any Pieris records at genus level (Unable to Verify) but add the following saved comment to the record:

Thanks for your record. Unfortunately I'm unable to verify the record as it is an 'unknown white' butterfly. However, this record will still provide Butterfly Conservation with information about the abundance of butterflies found in gardens & the record will be used in reporting.

If you had a large mixed goose flock of three species, what you would do is estimate how many of each you have and then record them using the 'enter a list of species' option, with a line for each, so for example

Canada Goose 100

Greylag Goose 350

Pink-footed Goose 50

Usually the life stage would be adult - you can add more specifics in the comments section. If there are young that had fledged that year I would include them in the count, if they are young goslings I would just count the adults and include the comment "family group with 5 goslings". 

I think it is unlikely that you would usually accurately assess the sex of geese in a large flock like that. If you had a single pair of geese there is the option to put 'mixed' in the sex option which would cover it.