En printempo floroj estas ĉie en la ĝardeno, kaj la vetero ŝajnas pli varma.

Yes. Both are possible.

• En printempo literally means in spring.
• Printempe means during spring / in springtime.

Both are natural Esperanto.
Using en + season noun is very straightforward for English speakers, while printempe uses the adverb-style ending -e that Esperanto often uses for times and seasons.

So:

• En printempo floroj estas ĉie...
• Printempe floroj estas ĉie...

Both mean essentially the same thing.

In Esperanto, -j marks the plural.

• floro = a flower
• floroj = flowers

So floroj estas ĉie means flowers are everywhere.

This is one of the most basic Esperanto endings:

• noun ending: -o
• plural ending: -j

Together: flor-o-j

Esperanto has the definite article la, but it is only used when you want to make something definite or specific.

Here, floroj means flowers in a general sense, not a specific previously mentioned group of flowers. So no article is needed.

Compare:

• Floroj estas ĉie en la ĝardeno. = Flowers are everywhere in the garden.
• La floroj estas ĉie en la ĝardeno. = The flowers are everywhere in the garden.

The second version would sound more like you are talking about particular flowers already known from context.

Estas is the verb to be, and here it simply states existence or presence in a place:

• floroj estas ĉie en la ĝardeno = flowers are everywhere in the garden

This is very natural Esperanto.

Of course, other verbs are possible if you want a different meaning:

• floroj kreskas ĉie = flowers grow everywhere
• floroj aperas ĉie = flowers appear everywhere

But estas is the simplest and most neutral choice for saying they are present all over the garden.

Ĉie means everywhere.

It is an adverb, so it describes location in a general way.

Examples:

• Ĉie estas homoj. = There are people everywhere.
• Mi serĉis ĉie. = I searched everywhere.

In your sentence, ĉie en la ĝardeno means everywhere in the garden.

Because en is the preposition meaning in.

The sentence has two separate location/time phrases:

• En printempo = in spring
• en la ĝardeno = in the garden

They are doing different jobs:

• the first tells when
• the second tells where

So it is completely normal to use en twice.

Because la ĝardeno refers to a specific garden, while floroj is more general.

• la ĝardeno = the garden
• floroj = flowers

In this sentence, the garden is treated as a definite place: the garden.
But the flowers are not being identified as a specific set already known to the listener.

This difference is very common in Esperanto, just as in English.

La vetero is the normal way to say the weather.

Even though English also says the weather, learners sometimes wonder whether Esperanto really needs the article here. Usually, yes: la vetero is the standard expression when talking about the weather in general or the weather in a given situation.

So:

• la vetero estas bela = the weather is nice
• la vetero ŝajnas pli varma = the weather seems warmer

Without la, it would sound less natural in most ordinary contexts.

Because ŝajnas is the conjugated verb that agrees with the subject la vetero.

• ŝajni = to seem
• ŝajnas = seems

Esperanto verbs do not change for person or number, but they do change for tense and mood:

• -as = present tense
• -is = past
• -os = future
• -us = conditional
• -u = imperative/jussive
• -i = infinitive

So:

• la vetero ŝajnas = the weather seems

Because varma is describing the noun vetero.

• varma = warm, as an adjective
• varme = warmly / in a warm way, as an adverb

Here the sentence means:

• the weather seems warmer

Since weather is a noun, the word describing it should be an adjective:

• la vetero ŝajnas pli varma

If you used varme, it would describe the manner of seeming, which is not what is meant.

Because there is no direct object here, and the location phrases are not showing motion toward something.

Key points:

1. Floroj is the subject, so no -n.
2. la vetero is also a subject in the second clause, so no -n.
3. en la ĝardeno shows location, not movement, so no -n.

Compare:

• Mi estas en la ĝardeno. = I am in the garden.
  
  • location only, no -n
• Mi iras en la ĝardenon. = I go into the garden.
  
  • movement toward, so ĝardenon

So in your sentence, no accusative is needed.

Because Esperanto can use pli without explicitly saying what the comparison is against, if that comparison is understood from context.

• pli varma = warmer
• pli varma ol hieraŭ = warmer than yesterday

In your sentence, the idea is simply that in spring, the weather seems warmer than before, than in another season, or than one might expect. The exact comparison is left unstated, just like in English when we say It seems warmer.

Yes, Esperanto word order is fairly flexible, as long as the sentence stays clear.

Original:

• En printempo floroj estas ĉie en la ĝardeno, kaj la vetero ŝajnas pli varma.

Possible variations:

• Floroj estas ĉie en la ĝardeno en printempo, kaj la vetero ŝajnas pli varma.
• En la ĝardeno floroj estas ĉie en printempo, kaj la vetero ŝajnas pli varma.

But the original order is natural because it starts with the time setting En printempo and then gives the description.

A rough guide:

• ĉie ≈ CHEE-eh
• ŝajnas ≈ SHAI-nahs

More exactly:

• ĉ sounds like ch in church
• ŝ sounds like sh in shoe
• aj sounds like eye
• stress in Esperanto is always on the second-to-last syllable

So:

• ĈI-e
• ŜAJ-nas

That regular stress pattern is very helpful in Esperanto pronunciation.