Лавинный датчик и налобный фонарь уже готовы.

Because the subject is actually two things joined by и: лавинный датчик + налобный фонарь.
In Russian, when two singular nouns are connected by и and together form the subject, the predicate usually goes in the plural.

So:

• датчик готов = the detector is ready
• фонарь готов = the headlamp is ready
• датчик и фонарь готовы = the detector and the headlamp are ready

Готовы is the short form of the adjective готовый.

Russian often uses short-form adjectives after a noun to mean something like a state or condition:

• фонарь готов = the headlamp is ready
• фонари готовы = the headlamps are ready

The full form готовые is usually used more like a normal attributive adjective before a noun:

• готовые устройства = ready devices

So in this sentence, готовы is correct because it means are ready.

In the present tense, Russian usually does not use a word for am / is / are.

So:

• фонарь готов literally looks like headlamp ready
• but it means the headlamp is ready

The verb to be is normally omitted in present-tense sentences like this.

They are in the nominative case, because they are the subject of the sentence.

You can tell because these are the basic dictionary forms:

• лавинный датчик
• налобный фонарь

Nothing in the sentence is forcing them into another case such as accusative, genitive, or instrumental.

Because both adjectives describe masculine singular nouns in the nominative case:

• датчик is masculine singular
• фонарь is masculine singular

So the adjectives agree with those nouns:

• лавинный датчик
• налобный фонарь

Russian adjectives must agree with the noun in gender, number, and case.

A common clue is the noun ending in the nominative singular:

• nouns ending in a consonant are often masculine: датчик
• many nouns ending in -ь can be masculine or feminine, so you learn them individually; фонарь is masculine

You can also see it from agreement:

• налобный фонарь uses masculine adjective form налобный
• if it were feminine, you would expect something like налобная

Уже means already.

It shows that the state has been reached by now:

• датчик и фонарь готовы = the detector and headlamp are ready
• датчик и фонарь уже готовы = the detector and headlamp are already ready

It often adds a sense like by this point or now.

Yes, Russian word order is flexible, and уже can sometimes move depending on emphasis.

For example:

• Лавинный датчик и налобный фонарь уже готовы.
• Уже лавинный датчик и налобный фонарь готовы.
• Лавинный датчик и налобный фонарь готовы уже.

The first version is the most neutral and natural here.
Other positions are possible, but they may sound more marked or stylistically different.

Because Russian, like English, can coordinate two noun phrases with и = and.

So:

• лавинный датчик и налобный фонарь = avalanche detector and headlamp

You do not need to repeat the adjective endings or the predicate separately unless you want extra emphasis.

Yes, but that would sound more emphatic or stylistically heavier:

• Лавинный датчик уже готов, и налобный фонарь уже готов.

This is grammatical, but less compact.
The original sentence is more natural when both items share the same idea of being ready.

Лавинный means related to avalanches. So лавинный датчик is an avalanche detector/beacon.

Лавиноопасный means avalanche-prone or dangerous because of avalanches, and it is used for places or conditions, for example:

• лавиноопасный склон = an avalanche-prone slope

So лавинный here is the correct adjective for equipment associated with avalanche safety.

More or less, yes.
It comes from лоб = forehead, so налобный means something like worn on the forehead or head-mounted.

So налобный фонарь is a headlamp / head torch.

This is a good example of Russian forming descriptive adjectives from body-part nouns.

• фонарь is a lamp/light device, often something portable or a flashlight-type object; in this phrase it means headlamp
• лампа is more generally a lamp
• свет means light as a phenomenon, not usually the device itself

So налобный фонарь is the natural expression for a headlamp, not налобная лампа in most standard usage.

In many outdoor and mountaineering contexts, yes.
It can refer to the device used in avalanche rescue, often called an avalanche transceiver or avalanche beacon in English.

A learner may see different English translations, but in Russian датчик here is the normal equipment term in context.

The predicate adjective would still agree with the whole plural subject, so it would still be plural if there were two items.

Examples:

• карта и аптечка уже готовы = map and first-aid kit are already ready
• устройство и приложение уже готовы = device and app are already ready

If there were only one noun, then the short-form adjective would match that noun:

• masculine: готов
• feminine: готова
• neuter: готово
• plural: готовы

A learner might want to know the pronunciation:

• лавИнный
• дАтчик
• налОбный
• фонАрь
• ужЕ
• готовЫ

Stress matters in Russian, so it is worth learning these together with the words.