Щенок радостно вилял хвостом и положил лапу мне на колено.

Because радостно is an adverb, and it modifies the verb phrase: it tells us how the puppy was wagging its tail.

• радостный щенок = a joyful puppy
• щенок радостно вилял хвостом = the puppy wagged its tail joyfully

A very common Russian pattern is:

• adjective: -ый / -ий / -ой
• matching adverb: often -о

So:

• радостный → радостно
• быстрый → быстро
• тихий → тихо

Because the verb вилять normally takes the thing being wagged in the instrumental case.

So Russian says:

• вилять хвостом = literally something like to wag with the tail

This is just the standard pattern, even though English uses a direct object:

• English: wag its tail
• Russian: вилять хвостом

You see similar patterns with some other body-part expressions too:

• махать рукой = to wave one’s hand
• пожать плечами = to shrug one’s shoulders

So хвостом is not random; it is the case the verb naturally requires.

Because лапу is the direct object of положил.

The basic verb pattern is:

• положить что? = to put/place what?

Here, what did the puppy place?
лапу = a paw / its paw

Since лапа is a feminine noun, its accusative singular form is:

• nominative: лапа
• accusative: лапу

So:

• положил лапу = put a paw

This is a very common Russian pattern with body parts, clothing, and things closely connected to a person.

мне на колено literally looks like to me onto the knee, but in natural English it means onto my knee.

Russian often avoids a possessive adjective when ownership is obvious. Compare:

• Он посмотрел мне в глаза. = He looked me in the eyes.
• Она взяла меня за руку. = She took me by the hand.
• Щенок положил лапу мне на колено. = The puppy put its paw on my knee.

You can say на моё колено, and it is understandable, but мне на колено is often more natural and idiomatic in this kind of context.

Because Russian uses different cases after на depending on whether you mean movement toward a place or location in a place.

• на + accusative = movement onto
• на + prepositional = location on

So here:

• положил лапу на колено = put a paw onto the knee
  movement → accusative
• лапа лежала на колене = the paw was lying on the knee
  location → prepositional

That is why the sentence has на колено.

Because the two verbs present the actions in different ways.

• вилял (imperfective) shows an ongoing or repeated action: the puppy was wagging its tail.
• положил (perfective) shows a single completed action with a result: it put its paw on the speaker’s knee.

This is very natural in Russian storytelling. One action can be background activity, while the next is a completed event.

So the sentence feels like:

• The puppy was happily wagging its tail and then put a paw on my knee.

If Russian used perfective for both, the feeling would be different.

That these verbs are in the past tense.

In Russian, the past tense is usually formed from the verb stem plus -л, and it also agrees with the subject in gender and number.

Here the subject is щенок, which is masculine singular, so we get:

• вилял
• положил

Compare:

• masculine: щенок положил
• feminine: собака положила
• plural: щенки положили

So the form tells you this is past tense, masculine singular.

Because both verbs have the same subject, and Russian normally does not repeat the subject if it is already clear.

So:

• Щенок радостно вилял хвостом и положил лапу мне на колено.

means:

• The puppy wagged its tail joyfully and put a paw on my knee.

Repeating щенок would usually sound unnecessary unless you wanted special emphasis or contrast.

Russian word order is fairly flexible because the case endings show the grammatical roles. But the sentence as given is a very natural, neutral order.

Current version:

• Щенок радостно вилял хвостом и положил лапу мне на колено.

That sounds like straightforward narration.

You can move parts around for emphasis, especially мне or радостно, but the tone changes. For example:

• Щенок радостно вилял хвостом и положил мне лапу на колено.

This is also natural, and many speakers might actually prefer this order.

So yes, word order is flexible, but not completely free: different orders sound more neutral, more emphatic, or more literary.

Yes—Russian stress is important and not always predictable. Useful stress marks here are:

• щено́к
• ра́достно
• виля́л
• хвосто́м
• положи́л
• ла́пу
• мне
• на коле́но

A few pronunciation notes:

• щ in щенок is a soft shch sound.
• вилял has stress on the last syllable.
• положил has stress on -жи-.
• колено has stress on -ле-: коле́но.

If you put the stress in the wrong place, Russian speakers will often still understand you, but correct stress makes you sound much more natural.

Yes, but it is usually unnecessary.

Russian often leaves out possessive words like свой, мой, его when the owner is obvious from context. Since a puppy naturally has its own paw, лапу already implies its paw.

So:

• положил лапу = natural
• положил свою лапу = possible, but more explicit or emphatic

Using свою might make sense if you wanted contrast, for example:

• Он положил свою лапу, а не чужую.
  He put his own paw, not someone else’s.

In ordinary narration, just лапу is the normal choice.