Старая калитка была открыта.

Because калитка is a feminine singular noun in the nominative case, and старая has to agree with it.

• старый = masculine singular
• старая = feminine singular
• старое = neuter singular
• старые = plural

So:

• старый дом = old house
• старая калитка = old gate

The ending changes to match the noun.

Калитка usually means a small gate, often the kind in a fence, garden wall, or yard entrance.

It is not quite the same as:

• дверь = door
• ворота = large gate(s), like a driveway gate
• калитка = a smaller pedestrian gate

So this sentence is talking about a gate/garden gate/wicket gate, not a house door.

Because была открыта describes the state of the gate: it was open.

• Старая калитка была открыта. = The old gate was open.
• Старая калитка открылась. = The old gate opened.

That second version focuses on the event/action of opening, while была открыта focuses on the resulting condition.

So in your sentence, the important idea is not that the gate opened, but that it stood in an open state.

Открыта is the short form; открытая is the full form.

In Russian, after быть and in similar predicate structures, the short form is very common when describing a condition or result:

• Калитка была открыта. = The gate was open.
• Окно закрыто. = The window is closed.
• Дверь заперта. = The door is locked.

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

• открытая калитка = an open gate

So:

• открытая калитка = open gate
• калитка была открыта = the gate was open

Usually, in a sentence like this, it most naturally means was open.

Russian была открыта can sometimes feel closer to a passive idea like had been opened, but in everyday usage it often simply describes the state:

• Дверь была открыта. = The door was open.

If you want to make the action clearer, Russian often uses another structure, for example:

• Калитку открыли. = They opened the gate.
• Калитка была открыта кем-то. = The gate was opened by someone.

Without an agent, English will often translate this as was open.

Because the sentence is in the past tense.

In Russian, the verb to be is usually omitted in the present tense, but it appears in the past and future.

Compare:

• Калитка открыта. = The gate is open.
• Калитка была открыта. = The gate was open.

So была is the past-tense form of быть for a feminine subject.

Because калитка is feminine.

Past-tense forms of быть agree with the subject’s gender and number:

• он был = he was
• она была = she was
• оно было = it was
• они были = they were

Since калитка is feminine, the sentence uses была.

Калитка is in the nominative singular, because it is the subject of the sentence.

Also:

• старая agrees with калитка, so it is nominative singular feminine too.
• открыта is a short-form predicate word, so it does not behave like a normal full adjective with case endings.

So the structure is basically:

• Старая калитка = subject
• была открыта = predicate

Historically, it comes from a passive participle related to открыть (to open), but in sentences like this it often behaves very much like a predicate adjective meaning open.

For a learner, the most practical way to understand it is:

• открытый = open, opened
• открыта = short form used in predicates

So in Калитка была открыта, you can think of открыта as the normal Russian way to say was open.

You can, but it does not sound as neutral or standard in this context.

• была открыта = the normal, concise way to say was open
• была открытой = more like was in an open state, with a slightly different stylistic feel

Russian usually prefers the short form for this kind of simple statement about a condition:

• Дверь была закрыта.
• Окно было открыто.

So for this sentence, была открыта is the best standard choice.

Yes. Russian word order is flexible, though the emphasis changes.

The neutral order is:

• Старая калитка была открыта.

You could also say:

• Калитка была открыта. = same basic meaning, without old
• Была открыта старая калитка. = more literary or marked; this puts more focus on the fact that it was the old gate that was open

Russian word order often reflects focus and emphasis, not just grammar.

Because Russian has no articles like a or the.

So Старая калитка была открыта could mean:

• The old gate was open
• An old gate was open

In real context, Russian speakers understand definiteness from the situation, previous sentences, or emphasis. English has to choose the or a, but Russian usually does not mark that directly.