Соль и сахар лежат на столе рядом с кружкой.

Russian commonly uses “position verbs” for location:

• лежать = to lie (resting horizontally / just “lying there” as an object)
• стоять = to stand (upright; e.g., a bottle, a glass)
• сидеть = to sit (often for people/animals, sometimes things like “a hat sits on someone’s head”)
• быть in the present tense is usually omitted (is/are is not normally said), so you typically choose a position verb instead: Соль и сахар лежат… = “Salt and sugar are lying (are placed)…”

Here salt and sugar (as items like a salt shaker and sugar bowl, or packets) are conceptualized as “lying” on the table.

Because the subject is compound: соль и сахар = “salt and sugar” → two things → plural verb:

• (они) лежат = “they lie/are lying”

If there were only one item:

• Соль лежит на столе. (singular лежит)

Each noun is singular, but the whole subject “A and B” is grammatically plural in Russian, just like in English:

• Salt and sugar are… (not is) So Russian uses plural agreement: соль и сахар лежат.

Because на changes case based on meaning:

• на + Accusative (на стол) = movement/direction “onto the table”
• на + Prepositional (на столе) = location “on the table”

Here it’s a static location, so на столе.

That’s the prepositional case used after на when it means location:

• стол (dictionary form, nominative)
• на столе (prepositional) = “on the table”

Many masculine nouns take -е in the prepositional: в доме, на столе, в городе, etc.

рядом (с) typically requires с + instrumental case:

• рядом с кем? с чем? = “next to whom/what?” So кружка becomes instrumental singular:
• кружка → кружкой And you include с: рядом с кружкой = “next to the mug.”

No—standard Russian needs с here: рядом с кружкой.
(You can drop с only in some colloquial or fixed expressions with other words, but with рядом you normally keep с.)

Because с (in this meaning “with/next to”) requires the instrumental case:

• кружка (nominative)
• кружку (accusative)
• кружке (prepositional/dative)
• кружкой (instrumental) ← needed after с in рядом с…

Not exactly:

• кружка = a mug (usually cylindrical, often larger, often for tea/coffee)
• чашка = a cup (often smaller, sometimes with a saucer)

Either could appear in similar sentences, but the meaning changes slightly.

Russian word order is flexible because case endings carry a lot of grammatical information.
Соль и сахар лежат на столе рядом с кружкой is neutral.

You can move parts for emphasis:

• На столе рядом с кружкой лежат соль и сахар. (focus on location; “On the table next to the mug, there are salt and sugar.”)
• Соль и сахар рядом с кружкой лежат на столе. (less neutral; can sound like you’re contrasting locations)

A natural question is:

• Где соль и сахар? = “Where are the salt and sugar?” Answer with the location:
• (Они) на столе рядом с кружкой. Or keep the full verb:
• Соль и сахар лежат на столе рядом с кружкой.

In Russian present tense, лежат can cover both:

• ongoing state (“are lying there now”)
• simple location (“are (located) on the table”)

Context usually decides; there’s no separate “continuous” form like English.

Yes, you’d likely switch to a verb of placement (perfective for a completed action), and use на стол (movement onto):

• (Кто-то) положил соль и сахар на стол рядом с кружкой. = “Someone put the salt and sugar on the table next to the mug.”

For an ongoing/unfinished action:

• кладёт (is putting) instead of положил.