Я часто думаю о ней.

Because думать in the sense of to think about someone/something normally uses о + the prepositional case:

• думать о ком? = to think about whom?
• думать о чём? = to think about what?

The pronoun она changes form after о, so:

• она → о ней

You cannot say о она, because Russian pronouns change form after prepositions.
You also do not use её here, because её is a different case form.

Ней is the prepositional case, singular, feminine.

In this sentence, the pattern is:

• о
  • prepositional

So о ней means about her.

A useful memory trick is to learn the question word with the preposition:

• о ком? = about whom?
• о чём? = about what?

So if you think думать о ком?, it becomes easier to remember why ней appears here.

This is a normal feature of Russian third-person pronouns after prepositions.

After many prepositions, Russian adds н- to forms of он / она / оно / они:

• к нему
• у неё
• с ними
• о ней

So ней is exactly what you should expect after a preposition like о.
This is something learners usually just have to get used to as a standard pattern.

Думаю is the 1st person singular form of думать.

So it means:

• я думаю = I think / I am thinking

The verb думать is imperfective, and here it is in the present tense.

A few forms are:

• я думаю
• ты думаешь
• он/она думает
• мы думаем
• вы думаете
• они думают

The ending -ю tells you the subject is I, which is one reason Russian can sometimes omit я.

Часто is an adverb, meaning often.

Adverbs in Russian do not change for:

• gender
• number
• case

So часто always stays часто.

That makes it easier than adjectives or pronouns, which often do change form.

Yes, it can change. Russian word order is more flexible than English.

Я часто думаю о ней is a very neutral and natural order.

Other versions are possible, for example:

• Я о ней часто думаю
• О ней я часто думаю

These all mean roughly the same thing, but the emphasis changes:

• Я часто думаю о ней = neutral statement
• Я о ней часто думаю = a bit more focus on о ней
• О ней я часто думаю = stronger emphasis on her

So the sentence you were given is probably the most straightforward, neutral version.

Yes, often you can.

Russian frequently drops subject pronouns when the verb ending already makes the subject clear. Since думаю already means I think, you can say:

• Часто думаю о ней.

This is natural if the context already makes it clear who is speaking.

Including я is also correct. It may sound a bit clearer, more explicit, or slightly more emphatic.

Yes, you can, and native speakers do say that.

However, there is a small difference in feel:

• думать о ней = more neutral, standard, and the safest choice
• думать про неё = more conversational or colloquial

Also notice the case change:

• о ней → prepositional
• про неё → accusative

So both are possible, but о ней is the more classic pattern learners should learn first.

Because о is the normal form before most consonants, and н is a consonant.

Russian uses:

• о before most consonants: о ней, о брате
• об before some vowel sounds: об этом
• обо in a few set expressions or for easier pronunciation: обо мне

So in this sentence, о ней is the correct form.

A simple learner-friendly pronunciation is:

• о ней ≈ a NYEY

A few helpful points:

• ней has the stress on ей
• the н is soft because of the following vowel sound
• in normal speech, о is usually unstressed and sounds weaker than a clear English oh

You do not pronounce it as two completely separate strong words. In natural speech, it flows together smoothly: о ней.