Short answers, pulled from the story.
The Gardens of Lucullus introduced Persian garden concepts to Europe around 60 BC. This space became a place of peace and tranquillity while holding deep religious meaning.
Most popular plants found in typical Roman family gardens included pine trees, roses, cypress, rosemary, and mulberry trees. Also possibly included dwarf trees pruned for ornamental purposes, tall trees, marigolds, hyacinths, narcissi, violets, saffron, cassia, thyme, lilies, irises, daisies, buttercups, geraniums, and olive chestnut almond citrus trees.
Lower class Romans used kitchen gardens as a source of food to provide for their families by growing herbs and vegetables within small plots. These gardens served practical needs across the Empire including Britain where species like garlic basil coriander dill parsley rosemary onion mustard grew.
Non-residential gardens referred to spaces either planted for public use or dedicated after owner's death such as temple gardens sacred groves parks attached to theaters or baths. Augustus constructed the Porticus Liviae a public garden on the Oppian Hill in Rome while other centers of wealth saw proliferation of these designs.
A free-standing Domus structure contained multiple rooms plus an atrium toward the front to collect rainwater known as impluvium. Toward the back of the house was often a hortus or peristylium open courtyard while villas served as country houses accessible from the city with maritime suburban and country types being most common.