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The image is quite blurry making it difficult to determine what is on your plant.

Neem Oil can be used to treat the leaves.

https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

Yes, these suckers should be cut off. They will take from the nutrients of the top of the tree. It will produce less of these as the condition of the main tree improves.

This article will help you on the care of these: https://www.gardeningknowhow.com/edible/fruits/olive/olive-tree-care-information.htm

The signs are consistent with water deficit (drought stress). I understand it's been a hot summer in SoCal. Check water needs with a soil moisture meter, don't guess.

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

I recommend that you re-pot the tree into a larger container. The present pot is too small to support the amount of foliage. It needs more soil volume for moisture reserves and root growth.

It sounds like you have done the right things. If mine I would spray the entire tree, foliar canopy and woody stems, now and a month later with a biological fungicide-bactericide. I'm not sure what is available in Italy, but in the US I use this:

http://certisusa.com/pest_management_products/biofungicides/doublenickel_LC_fungicide.htm

https://www.domyown.com/monterey-complete-disease-control-readytouse-p-17485.html

http://ipm.ucanr.edu/PMG/r583100411.html

Growing olives in zone 7 will be challenging but doable if you are willing to take steps to protect it in cold weather. Start looking at record cold temperatures in a city near you to give you an idea of what you might be facing and draft a protection strategy that will keep the tree's air temperature above 20 F. Long spells at 20 F, more than several hours, will also damage many olive trees. I've provided links to 2 articles. One discusses the more cold tolerant olive varieties. The second one gives reasons why olives can't be grown in zone 7. It seems that it can be done if provide adequate protection. Now that many of us have experienced a Polar Vortex and that days of spring-like air come mid-winter, winter temps seem less predictable. This wrecks havoc with any tree that flowers in spring.
https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-7/zone-7-olive-trees.htm
https://olivetreegrowers.com/blog.php?view=detail&id=114