Зимой я всегда кладу крем для рук в сумочку.

In Russian, names of seasons are often used in the instrumental case to express during that season:

• зимой = in winter
• летом = in summer
• осенью = in autumn
• весной = in spring

So зимой is a very common adverb-like way to say in winter.
You can also say в зимнее время or similar expressions, but зимой is the normal everyday choice.

Кладу is the 1st person singular present tense form of класть.

• класть = to put, to place
• я кладу = I put / I am putting

This sentence describes a habitual action, and Russian normally uses the imperfective verb for repeated or usual actions. Since the sentence means something like I always put hand cream in my purse in winter, кладу is exactly the right choice.

Because положу is perfective and usually refers to a single completed future action:

• я положу = I will put

But this sentence is about something the speaker regularly does, not one future event. For repeated habits, Russian prefers the imperfective:

• я всегда кладу = I always put

So кладу fits always much better than положу.

Крем is in the accusative case because it is the direct object of кладу: it is the thing being put somewhere.

For inanimate masculine singular nouns, the accusative is usually the same as the nominative:

• nominative: крем
• accusative: крем

So the form does not change, even though the case has changed.

Russian usually expresses for with the preposition для, which requires the genitive case.

• для = for
• руки = hands
• genitive plural: рук

So:

• крем для рук = cream for hands = hand cream

This is the normal Russian way to say hand cream.

Because the phrase literally means cream for hands, and in Russian that idea is normally expressed with the plural:

• руки = hands
• для рук = for hands

Even though English often uses a singular noun in compounds like hand cream, Russian commonly uses a fuller phrase with для + plural noun.

Because в can mean either:

• into a place → usually takes the accusative
• in a place → usually takes the prepositional

Here the cream is being put into the purse, so Russian uses the accusative:

• в сумочку = into the purse / into the handbag

Compare:

• Кладу крем в сумочку. = I put the cream into the purse.
• Крем лежит в сумочке. = The cream is in the purse.

Сумочка is a diminutive form of сумка.

• сумка = bag
• сумочка = little bag, purse, handbag

In many contexts, сумочка sounds natural when talking about a woman's handbag or purse. It can sound a bit softer, more personal, or more specific than сумка.

So both are possible depending on context, but сумочку works well here.

Yes. Russian word order is fairly flexible because case endings show the grammatical roles.

This sentence could also be phrased as:

• Зимой я всегда кладу в сумочку крем для рук.
• Я зимой всегда кладу крем для рук в сумочку.

These versions mean basically the same thing, but the emphasis shifts slightly. The original sentence is perfectly natural. Putting зимой first is very normal because it sets the time frame right away.

Not always. Russian often omits subject pronouns when they are clear from the verb ending.

• кладу already means I put

So you could say:

• Зимой всегда кладу крем для рук в сумочку.

That is grammatical and natural.
However, including я is also completely fine. It can make the sentence a bit clearer or slightly more explicit.

Всегда means always, so it strongly suggests a habitual, repeated action.

That is why the present-tense imperfective verb кладу is used. In Russian, the present tense can describe what someone regularly does, not only what they are doing right now.

So:

• я всегда кладу = I always put / I always keep putting

In natural English, this is often translated simply as I always put.