Lengthy this one. Maybe should have been broken down into little subquestions.
It's actually a bit simplistic to tie cultures entirely to their biomes; people move around, groups migrate and what started in one biome might end up in another, cultural and socio-religious-economic-political reasons might advocate certain approaches. Given how culturally complex we all are, it can't really be used to "classify" any particular group of people or explain completely why they do things the way they do.
Also some needs are common across all human groups and the response to these can be shared across all of us. Eg. Community/family ties are generally very important to people who are highly dependent on the vagaries of nature, no matter whether they live in the tundra or in the rainforest.
So let me do this merely by describing the environment and the limitations it would pose and how people got around these historically.
Ice-caps/tundra - Extreme cold and ice cover mean it's highly inhospitable to plant life. The only food sources available would be animal-based obtained from either fishing, hunting or herding (seals, reindeer, sea-life), all of which would require an experienced and travel-hardened set of people. Genetic adaptations in some native peoples give them greater tolerance for a diet completely based off animal foods. Herders would need to keep moving to give their animals enough to graze on in the sparse grass-lichen-moss environment. Ice and permafrost move which means you really couldn't have tall permanent buildings with associated deep foundations. People would need cold-resistant clothing and shelter and a means of obtaining heat (very difficult to start fires and keep them going unless you are aided by modern tools like stoves/gas lighters etc). The cold makes excellent storage and food could be preserved for times of need.
Temperate grasslands - Very hot summers, very cold winters, some precipitation in spring, not enough to reach the deeper layers of soil. Mostly all grass. Prone to drought and quick-burning fires. No cover really from strong arctic winds. Not many trees, no firewood. Need to find other fuel for fire. Some groups traditionally herd animals (plenty of grass) for food and dung for fuel. Soil is very fertile from the decay of grass over generations. Some people near assured water supply grow grass-based, cold resistant crops, eg wheat. Low diversity of animals but very plentiful: bison, deer, caribou: can mean good hunting. Overhunting, overgrazing and overcropping have all happened before around the world.
Tropical grasslands - Warm and hot climate with plenty of rain concentrated in 6-8 months in a year with drought the rest of the time with possible annual fires. Could be formed due to human/elephant clearing. Stands of deep-rooted trees and forests in between. A very large range of wildlife, both herbivores and predators. Any people here would need to worry about safety from animals and water supply in the dry season. Food can be obtained from hunting and herding, from plant/tree sources and from agriculture.
Hilly regions - as you have described. Crops of fruits, tea and foodgrains are possible with terraced fields at lower altitudes where rainfall is moderate and soil well drained but not overdrained.
Alpine regions - as above with the added problem of low oxygen levels at high altitudes, barren at very high altitudes, vegetation limited to small specialized plants, grasses and mosses at high altitudes, evergreens at lower altitudes. Some groups have genetic adaptations for efficient processing of oxygen at high altitudes.
Plateau/interior regions - Very hot and dry for most of the year, with moisture obtained from seasonal rain, lakes or flowing rivers. Drought resistant crops can survive well. Temperature swings can be marked, with rapid cooling at night.
Coastal plains/islands - Well precipitated and close to marine ecosystem. Agriculture and fishing usually well developed. People here could be very good mariners and navigators. Exposed to storms, sea-level changes, tsunamis and other extreme coastal-events.
Temperate forests - Well precipitated at places, support a varied animal life. Plenty of timber for construction.
Tropical forests/ Rainforests - Thick forest cover. Very warm and humid. People here couldn't really store food very long, they'd just have to find food seasonally and consume it. Also forest soil can be poor, meaning hunting would be a part of life with or without small plots of cleared crop lands with very small harvests. Some weapons such as blowpipes can be more efficient than bows and arrows at short ranges especially upwards into the canopy. These can be supplemented by easily found plant/animal toxins for more effectiveness. Pests such as mosquitoes plentiful.
Swampy/marshy places - Very poor soil; water-logging drains essential nutrients. Plant species growing here would look lush and green but may not provide much nutrition. Not really agricultural land but some groups have crop plant species that are tolerant to marshy/briny/brackish conditions (eg. wild marsh rice) - these efforts would be on a small local scale. Also some plant foods are still found along with an abundance of aquatic/bird life. The moisture and heat would encourage bacterial/pest (esp mosquitoes) growth. Some human groups have evolved malarial resistance as response.
Deserts (excluding Antarctica) - Hot during day, cold at night. Very little precipitation and very, very dry. Water key to life here. Some vegetation possible in areas depending on how close the water table is to the surface. Soil can be surprisingly good in these isolated oasis areas, with vine-based plants such as melons, pomegranates and some vegetables and dry condition resistant trees such as date palms and succulent cactii. Sand areas and rocky plateau areas would pose travel/transport problems due to heat, dry loose fine dust/sand and lack of water. Shifting sand always a problem for cultivated areas.
Nutrient starved places - usually a monoculture in place as far as vegetation is concerned, eg only a certain type of plants around such as grass or moss. Ecosystem geared around this monoculture.
Please feel free to add more.