Ядовитый мухомор растёт под деревом.

Because it agrees with мухомор.

Мухомор is:

• masculine
• singular
• nominative

So the adjective must also be masculine, singular, nominative:

• ядовитый мухомор

If the noun changed, the adjective would change too:

• ядовитая ягода — feminine
• ядовитое растение — neuter
• ядовитые грибы — plural

It is in the nominative singular.

That is the normal case for the subject of a sentence, the thing that performs the action or is in the state described by the verb. Here, мухомор is the thing that растёт.

Расти is the infinitive, meaning to grow.

In a full sentence, Russian usually needs a finite verb form, not the infinitive.
Растёт is the 3rd person singular present form of расти.

So:

• расти = to grow
• растёт = grows / is growing

It is singular because the subject мухомор is singular.

It can mean either, depending on context.

Russian present tense often covers both:

• the general meaning grows
• the ongoing meaning is growing

So this sentence could describe:

• a general fact
• something happening right now

Russian does not usually need a separate form like English is growing.

The letter ё shows a stressed yo sound.

So растёт is pronounced roughly ras-TYOT.

A useful thing to know: in many Russian texts, especially informal ones, people write е instead of ё. So you may often see:

• растет

But it is still pronounced растёт.

Because под can take different cases depending on the meaning.

Here it means location, under in the sense of where something is. With that meaning, Russian uses the instrumental:

• под деревом = under the tree

If there is movement toward a position under something, Russian often uses the accusative:

• положить под дерево = to put something under the tree

So:

• под деревом = location
• под дерево = motion toward that location

Деревом is the instrumental singular of дерево.

The noun дерево is neuter and ends in -о.
A common instrumental singular ending for this type is -ом:

• дерево → деревом

It is in the instrumental here because of под with the meaning of location.

Russian has no articles.

So мухомор can mean:

• a fly agaric
• the fly agaric

And под деревом can mean:

• under a tree
• under the tree

Context tells you which English article makes the most sense.

No, Russian word order is more flexible than English word order.

The sentence:

• Ядовитый мухомор растёт под деревом

is a neutral, natural way to say it.

But Russian can move words around for emphasis, because the endings already show the grammatical roles. For example:

• Под деревом растёт ядовитый мухомор
  This gives more emphasis to the location.

The basic meaning stays similar, but the focus changes.

The standard stress pattern is:

• ядови́тый
• мухомо́р
• растёт
• де́ревом

So the full sentence is:

Ядови́тый мухомо́р растёт под де́ревом.

Stress is very important in Russian because it is not always predictable, and it can affect vowel pronunciation.

Because Russian normally does not use a present-tense form of to be in sentences like this.

In English, you might expect something like is under a tree or is poisonous, but in Russian the present-tense to be is usually omitted.

In this sentence, the only verb needed is растёт. Russian does not need an extra present-tense is here.