How to model a database structure for exam dates and date ranges

What you are proposing is a good solution for your requirement of M:N products to categories and hierarchical categories.

To avoid exposing yourself to numerous updates: You need to do two things to ensure that you don't have a lot of updates in your intersection table.

First, you need to be sure that your categories have a stable, persistent primary key.

Second, you need to link food items to leaf categories. Don't join Cherry to Red, Healthy, Fruit and Food - just join it to Red and Healthy. Your nested sets take care of all of the secondary (and higher level) associations.

Categories and Buckets.

create table categories
(
  category_id int primary key
);

create table buckets
(
  bucket_id int primary key
);

Depending on how to interpret "an item may belong to at most 1 bucket" you need to do some different things with Items.

An item has to belong to one bucket.

create table items
(
  item_id int primary key,
  category_id int not null references categories(category_id),
  bucket_id int not null references buckets(bucket_id),
  unique (category_id, bucket_id)
);

If an item may belong to one bucket but does not have to you need to allow for null values in bucket_id and use a filtered index as the unique constraint.

create table items
(
  item_id int primary key,
  category_id int not null references categories(category_id),
  bucket_id int references buckets(bucket_id)
);

create unique index uq_items on items(category_id, bucket_id) where bucket_id is not null;

There is a collection of items where each item belongs to 1 category.

category_id as a foreign key in items.

an item may belong to at most 1 bucket.

bucket_id as a foreign key in items.

There is a collection of buckets which may contain at most 1 item from each category.

A unique constraint on bucket_id and category_id in items.