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Lemons will, commonly, drop smaller fruit if there are too many on the tree. This is so that they can focus growth on the remaining fruits. This is no reason for concern. I'm sorry but you cannot plant those, as they don't contain mature seeds.

https://www.gardngro.co.nz/shop/Fertilisers/Citrus+Fertiliser+3kg.html

https://www.tuigarden.co.nz/product/tui-citrus-food/

There are more, I just searched Google for "citrus fertilizer nz"

Yours is probaly a "lemonade lemon tree", a hybrid between lemon and mandarin orange.

You need to protect it from the frost. It can take the full sun but not temperatures below freezing.
And water it before it dries out completely, it may be drought stressed from water deficit as well as frozen.
From the photo it does not appear dead. You can save it with better care.

Yes you can prune off the driest, possibly dead branches, but don't prune off any live green growth now, it needs that for recovery.

https://homeguides.sfgate.com/care-lemonade-lemon-citrus-trees-57365.html

https://www.gardeningknowhow.com/edible/fruits/lemons/growing-lemon-trees-containers.htm

There are a few things that can cause this. Sometimes the tree gets too crowded with its own growth that the fruit does not get enough light through the canopy. It could also be a watering issue. Here is a link to more information:

https://www.gardeningknowhow.com/edible/fruits/lemons/lemons-stay-green.htm

A good rich potting soil, adequate water and fertilization are the first considerations.

If the plant has been in the existing pot more than a year or two, consider re-potting into a larger pot with good potting soil like this:
https://www.amazon.co.uk/COMPO-SANA®-Citrus-Potting-Soil/dp/B001B1NVHG/ref=sr_1_fkmr0_1?ie=UTF8&qid=1538486571&sr=8-1-fkmr0&keywords=down+to+earth+potting+soil

If it's a new planting this year or last, and you think the soil is as good as you can get, then the next issue is water. Too little or too much can be a problem.

Consider using a soil moisture meter to take the guesswork out of when to water. Don't under water or over water, excess water can cause root rot and the signs you describe. Too little water can cause drought stress, loss of leaves and branch die-back.

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm

Fertilize at least spring and fall with a slow release natural fertilizer like this:

https://www.amazon.co.uk/Down-Earth-Organic-Fertilizer-25-Pound/dp/B00KO4KTRG

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