Learn English – Adjective or verb

It depends on the context; its position in the sentence. 1) If the word with ing comes as a subject or an object, it's a noun. eg. Smiling is an act of kindness. 2) If it comes after the auxiliary "be" it's a verb. eg. The girl is smiling. 3) If it comes before a noun, describing that noun, then it's an adjective. eg. Smiling faces are lovely.
Some words with ing are always adjectives like 'interesting' and 'exciting'.

The verb in the example is "to paint".  The principle parts of this verb are "paint", "painted" and "painted".  The second and third principle parts happen to be the same, as they are for most English verbs.

However, that similarity does not occur in every English verb.  For example, the principle parts of "to take" are "take", "took", "taken".
 

• She painted this picture.
• She took this picture.

This sentence uses the second principle part -- the past tense form.  The word is used as a verb, has a tense, and has no auxiliary. 
 

• She has painted this picture.
• She has taken this picture.

This sentence uses the third principle part -- the participle form.  The word is used as part of a complete verb that includes the auxiliary "has".  The complete verb is in the present tense and the perfect aspect.

The version that uses the verb "to take" cannot be mistaken for the past tense form -- "taken" looks and sounds nothing like "took".  For the version that uses "to paint", it is only the fact that there is an auxiliary verb that indicates "painted" is the participle form.

By the way, this is not a passive voice construction.  Passive voice uses the auxiliary "to be" instead of "to have".  For example, "the picture was taken" or "the picture will be painted".
 

Participles can be used as a part of a complete verb, but they can also be used as modifiers.  When they are modifiers, they are not combined with auxiliaries.

In general, participles that are alone come before the noun, as in "the painted picture".  Participial phrases, however, tend to follow the noun:

• The picture painted by Karen is now in a museum.
• The picture taken by Karen is now in a museum.

The phrase "painted by Karen" does not have a tense, does not form a predicate, and does not have a subject.  Instead, it behaves like an adjective and answers the question "which picture?"  There is a verb in this sentence, but that verb is "is".  The subject of "is" is "picture".

Prepositional phrases can do the same sort of job. Consider:

• The picture of three apples is now in a museum.

   

Your comment includes an example sentence that has bad grammar:

• The picture was painted by Karen is now in a museum.

Here's the problem:  "Was painted" is a complete verb.  It has a tense, it forms a predicate, and it needs a subject.  "Is" is also a complete verb, possessing tense and forming a predicate and needing a subject.  Unfortunately, there's only one subject available, and it can't satisfy both predicates.

There are a few ways to fix that problem. One way is to remove "was" and let "painted by Karen" act as a modifier. Another way is to give "was painted by Karen" its own subject, such as "The picture that was painted by Karen is now in a museum." Yet another way is to join the two predicates with a conjunction, so that the one subject can satisfy the resulting compound: "The picture was painted by Karen and is now in a museum."