אולי השלט על השולחן; תבדקי אם הכפתור האדום עובד.

In Hebrew, the verb to be is usually omitted in the present tense. So השלט על השולחן literally looks like the remote on the table, but it means the remote is on the table.

This is very normal Hebrew grammar, not an incomplete sentence.

אולי means maybe or perhaps. It often comes at the beginning of the clause, just like English maybe:

אולי השלט על השולחן = Maybe the remote is on the table

It can appear in other positions too, but sentence-initial position is very common and natural.

Yes. שלט can mean remote control or sign, depending on context.

Here, it means the remote. The prefix ה־ is the definite article, so:

שלט = a remote / sign
השלט = the remote / the sign

Context tells you which meaning is intended.

על is the preposition on. So:

על השולחן = on the table

Because Hebrew usually drops present-tense is, השלט על השולחן is the normal way to say the remote is on the table.

Grammatically, תבדקי is the second person feminine singular future form of לבדוק (to check): you will check.

But in everyday spoken Hebrew, future forms are very often used to give commands, especially in casual speech. So here תבדקי means:

check / you should check

It is being said to one female.

Yes. The formal imperative to one woman would be בדקי.

So both can mean check! when speaking to a woman:

בדקי = formal/textbook imperative
תבדקי = very common spoken Hebrew command style

In modern everyday Hebrew, the future-form command is often more common in conversation.

The form changes according to gender and number.

From לבדוק:

To one man: תבדוק or formal בדוק
To one woman: תבדקי or formal בדקי
To more than one person: תבדקו or formal בדקו

So תבדקי specifically tells you the speaker is addressing one female.

Here אם introduces an indirect yes/no question, so in English it is often best translated as whether:

תבדקי אם הכפתור האדום עובד = Check whether the red button works

In many cases, English also allows if, and Hebrew uses אם for both ideas.

In Hebrew, when a noun is definite, an adjective describing it is also usually definite.

So:

כפתור אדום = a red button
הכפתור האדום = the red button

This is a very important pattern in Hebrew: the adjective agrees with the noun in definiteness, as well as in gender and number.

Because that is the normal order in Hebrew. Most adjectives come after the noun:

כפתור אדום = red button
שולחן גדול = big table
ילד חכם = smart boy

So הכפתור האדום literally follows the Hebrew pattern the button the-red.

עובד is the masculine singular present-tense form of לעבוד, which usually means to work. When talking about a device or button, it means works / functions.

It is masculine singular because כפתור is masculine singular.

So:

הכפתור האדום עובד = the red button works

If the noun were feminine, you would use עובדת.

The semicolon separates two closely related ideas:

אולי השלט על השולחן
תבדקי אם הכפתור האדום עובד

The first part gives a possibility, and the second part gives an instruction. A semicolon is one way to join them neatly. In less formal writing, you might also see a comma or a period instead.