Держи меня в курсе. Будем на связи. Передавай привет.

Yes. All three are addressed to one person in the informal ты form.

Informal:

• Держи меня в курсе.
• Будем на связи.
• Передавай привет.

Polite / plural:

• Держите меня в курсе.
• Будем на связи.
  This one usually stays the same, because it means We’ll be in touch.
• Передавайте привет.

So the main changes are:

• держи → держите
• передавай → передавайте

Держи is the imperative form of держать (to hold / keep).

In this expression, держать кого-то в курсе means to keep someone informed.

So:

• держать = infinitive
• держи = imperative, informal singular: keep
• держите = imperative, polite/plural: keep

It is not being used in the literal sense of physically holding someone. It is part of the idiom держать в курсе.

Because меня is the direct object of держи.

The pattern is:

• держать кого? в курсе
• keep whom? informed

So:

• держи меня в курсе = keep me informed
• держи его в курсе = keep him informed
• держите нас в курсе = keep us informed

Even though English says keep me in the loop, Russian uses a structure where the person is treated as the object: keep me in the know.

В курсе is a very common expression meaning aware, informed, or up to date.

Examples:

• Я в курсе. = I know / I’m aware / I’m up to date.
• Ты в курсе? = Do you know? / Are you aware?
• Держи меня в курсе. = Keep me informed.

Learners often try to understand курс literally, but here it is best learned as a fixed phrase:

• быть в курсе = to be in the know
• держать кого-то в курсе = to keep someone informed

Yes, very naturally.

Common expansions:

• Держи меня в курсе событий. = Keep me updated on what’s happening.
• Держи меня в курсе дела. = Keep me informed about the matter.
• Держи меня в курсе, как всё пройдёт. = Keep me posted on how it goes.

Without extra words, Держи меня в курсе already sounds complete and natural.

Because this expression literally means We will be in touch, and that is exactly how Russian often says it.

• будем = we will be
• на связи = in contact / reachable / connected

So it is not a command like stay in touch. It is a statement about future contact:

• Будем на связи. = We’ll be in touch.

You can say:

• Будь на связи. = Stay reachable / stay in touch.
• Будьте на связи. = polite/plural version

But that has a slightly different feel: more like an instruction or request.
Будем на связи is a very common friendly sign-off.

It is the prepositional singular of связь.

• nominative: связь
• prepositional: о связи / на связи

After на meaning location/state, Russian often uses the prepositional case:

• на работе = at work
• на месте = in place
• на связи = in contact / on the line

In modern Russian, на связи is basically a fixed expression.

Literally, it is something like on the connection / on the line.

It comes from the idea of communication being open or available. Depending on context, it can mean:

• in touch
• reachable
• online / available
• on the line (for calls/radio communication)

Examples:

• Я на связи. = I’m available / I’m reachable / I’m online.
• Оставайся на связи. = Stay in touch / Stay reachable.
• Будем на связи. = We’ll be in touch.

Both are possible, but they are slightly different in aspect and tone.

• передавай = imperfective imperative
• передай = perfective imperative

Very roughly:

• Передавай привет.
  Sounds more like a general, friendly formula: Say hi / Give my regards.
• Передай привет.
  Sounds more like one конкретный action: Make sure to pass on my regards.

In everyday speech, both are common. In a closing like this, Передавай привет sounds especially natural and conversational.

That is normal. Russian often leaves the recipient understood from context.

• Передавай привет. = Say hi / Give my regards.
• Передавай привет маме. = Say hi to your mom.
• Передавай всем привет. = Say hi to everyone.
• Передавай от меня привет. = Give my regards from me.

If the person or group is obvious from the conversation, Russian can omit it.

Usually the person receiving the greeting goes in the dative case.

Examples:

• Передавай привет брату. = Say hi to your brother.
• Передавайте привет родителям. = Give my regards to your parents.
• Передай Маше привет. = Say hi to Masha.

So the pattern is:

• передавать / передать привет кому?

That кому? question tells you it takes the dative.

Yes. They sound like a very natural conversational sign-off, especially in messages, phone calls, or casual goodbyes.

Together they give the sense of:

• Keep me posted.
• We’ll stay in touch.
• Say hi for me.

That kind of cluster is very typical in spoken Russian and friendly texting.

The word order is neutral and natural.

• Держи меня в курсе is the standard order.
• Будем на связи is the standard expression.
• Передавай привет is also standard.

Russian word order is flexible, but changing it here would usually either sound marked or just less natural.

For example:

• Меня держи в курсе is possible, but it adds emphasis to me.
• Привет передавай is also possible with special emphasis, but not the most neutral default.

So for a learner, the original order is exactly the one worth memorizing.

Like this:

• держи́ меня́ в ку́рсе
• бу́дем на связи́
• передава́й приве́т

A couple of useful notes:

• держи́ has stress on the last syllable
• ку́рсе has stress on ку́р-
• связи́ has stress on the last syllable
• передава́й has stress on -ва́й
• приве́т has stress on the second syllable

Getting the stress right will make these phrases sound much more natural.