Мне досадно, когда я забываю проездной дома.

Because досадно is used in an impersonal construction.

In Russian, feelings like мне грустно, ему холодно, нам обидно, мне досадно are often expressed as:

• [person in the dative] + [state word]

So:

• мне = to me / for me
• досадно = annoying / vexing / frustrating

Literally, Мне досадно is something like It is vexing to me.

English usually says I’m annoyed or It annoys me, but Russian often uses this dative pattern instead of я.

Not exactly. Досадно means something like:

• annoying
• frustrating
• vexing
• irritating in a mildly regretful way

It is often used when something unpleasant happens because of a mistake, bad luck, or inconvenience.

Compare:

• грустно = sad
• обидно = hurt / offensive / disappointing in a way that feels unfair
• досадно = annoying, frustrating

So if you forget your travel pass at home, досадно fits well because it is not deep sadness; it is more like Ugh, how annoying.

In the present tense, Russian usually omits the verb to be.

So:

• Мне досадно literally has no spoken/written is
• but it means It is annoying to me / I’m annoyed

This is normal in Russian:

• Мне холодно = I am cold
• Ему интересно = He is interested / It is interesting to him
• Нам весело = We are having fun

If you wanted past or future, a form of быть would appear:

• Мне было досадно = I was annoyed
• Мне будет досадно = I will be annoyed

Забываю is the imperfective verb, and here it suggests a repeated or general situation.

The sentence means something like:

• It annoys me when I forget my travel pass at home
• I find it frustrating whenever I leave my travel pass at home

So this is not about one specific event. It is about a type of situation that happens from time to time.

Compare:

• когда я забываю... = when(ever) I forget..., in general / on those occasions
• когда я забыл... = when I forgot..., referring to a specific past event
• когда я забуду... = when I forget..., referring to a future occasion

In this sentence, imperfective is the natural choice because it describes a habitual or repeated source of annoyance.

Here проездной is in the accusative singular, as the direct object of забываю.

The word is short for something like:

• проездной билет = travel pass / transit pass

In everyday Russian, people often just say проездной as a noun by itself.

Since забывать takes a direct object, you use the accusative:

• забываю что? → проездной

For this masculine inanimate noun, the accusative looks the same as the nominative:

• nominative: проездной
• accusative: проездной

So the form does not change, but the case function is accusative.

Because дома here means at home.

This is a very common adverb-like word in Russian:

• дома = at home
• домой = homeward / to home

Compare:

• Я забыл проездной дома. = I forgot my travel pass at home.
• Я иду домой. = I am going home.

В доме means in the house/building, which is more literal and less natural if you simply mean at home.

So:

• дома = location
• домой = direction

Here когда is best understood as when(ever).

Because the main sentence describes a general emotional reaction, когда does not point to one specific moment only. It means something like:

• when I forget...
• whenever I forget...

Russian often uses когда in this broader sense without needing a separate word for whenever.

So the sentence is not necessarily about one event; it expresses a recurring pattern: It annoys me whenever I forget my travel pass at home.

You could, but it would sound different and less natural in this context.

• Мне досадно = I feel annoyed / frustrated
• Я досадую = I am vexed / I feel vexation

The verb досадовать exists, but it is less common in everyday conversational Russian than the impersonal construction with досадно.

For natural modern speech, Мне досадно, когда... sounds much more idiomatic.

Yes. Russian word order is flexible, and the basic meaning would stay the same.

Possible variants:

• Мне досадно, когда я забываю проездной дома.
• Когда я забываю проездной дома, мне досадно.
• Мне досадно, когда дома я забываю проездной.
  This is possible, but less neutral.

The version you were given is the most neutral and natural.

In general:

• first position often sets the emotional frame: Мне досадно
• then the cause is explained: когда я забываю проездной дома

In Russian, забыть что-то дома often means to leave something behind at home by forgetting it.

So although the verb is literally forget, the whole phrase naturally corresponds to English:

• I forget my travel pass at home
• I leave my travel pass at home

In English, leave behind is often more idiomatic, but Russian commonly uses забыть for this situation.

You would add the appropriate form of быть to the досадно construction, and you could also adjust the verb in the когда-clause if needed.

Examples:

• Мне было досадно, когда я забыл проездной дома.
  I was annoyed when I forgot/left my travel pass at home.

• Мне будет досадно, если я забуду проездной дома.
  I’ll be annoyed if I forget/leave my travel pass at home.

Notice that in the future, Russian often prefers если (if) if the event is only possible, rather than когда if it has not happened yet.

So:

• когда = when / whenever
• если = if

That is why a future version often changes more than just the tense.