בביקורת הביטחון ביקשו ממני לפתוח את התיק ולשים את המחשב על השולחן.

Because it is really two parts merged together:

• ב־ = in / at
• הביקורת = the inspection / the check

In Hebrew, ב + ה usually contracts, so:

• ב + הביקורת → בביקורת

So בביקורת הביטחון means at the security check / during the security inspection.

This is a construct phrase (called סמיכות in Hebrew), where one noun is linked to another:

• ביקורת = inspection/check
• הביטחון = the security

Together, ביקורת הביטחון means the security check or security inspection.

A useful thing to notice: in this structure, the first noun usually does not take ה־, even when the whole phrase is definite. The definiteness is shown on the second noun:

• ביקורת הביטחון = the security check

Not הביקורת הביטחון.

ביקשו literally means they asked / requested.

In Hebrew, just like in English, the plural can be used when the exact person is not important or not named:

• ביקשו ממני... = they asked me...
• In natural English, this often means someone at security asked me...

So even if only one security officer spoke, Hebrew may still use the plural in a general, impersonal way.

ממני means from me.

It comes from the preposition מ־ / מן (from) plus אני (I / me):

• מ־ + אני → ממני

After the verb לבקש (to ask / request), Hebrew often uses מ־ to mark the person being asked:

• ביקשו ממני לפתוח...
  = They asked me to open... = literally something like They requested from me to open...

This structure is very common in Hebrew.

They are infinitives:

• לפתוח = to open
• לשים = to put

After ביקשו ממני (they asked me), Hebrew uses an infinitive to say what the person was asked to do:

• ביקשו ממני לפתוח את התיק = They asked me to open the bag
• ולשים את המחשב... = and to put the computer...

The ל־ here is the normal marker of the infinitive, like English to in to open, to put.

את marks a definite direct object.

It does not have a separate English translation here, but it tells you that the noun after it is the direct object of the verb and is definite.

So:

• לפתוח את התיק = to open the bag
• לשים את המחשב = to put the computer

You use את because:

• התיק = the bag
• המחשב = the computer

Both are definite nouns.

Because השולחן is not the direct object of the verb.

In:

• לשים את המחשב על השולחן

the direct object is:

• את המחשב = the computer

But על השולחן is a prepositional phrase:

• על = on
• השולחן = the table

So השולחן is part of on the table, not the direct object. That is why there is no את there.

Yes. לשים is the very common everyday verb for to put / to place / to set.

So:

• לשים את המחשב על השולחן = to put the computer on the table

You may also encounter להניח, which can also mean to place / set down, sometimes sounding a little more formal or specific depending on context. But לשים is extremely common and natural in everyday speech.

Because the sentence refers to specific, known items in the situation:

• התיק = the bag (your bag)
• המחשב = the computer (your computer)
• השולחן = the table (the table there at security)

In Hebrew, as in English, it is very natural to use the when both speaker and listener know which object is meant from the context.

The sentence order is very natural Hebrew:

• בביקורת הביטחון = setting/context first
• ביקשו ממני = what happened
• לפתוח את התיק ולשים את המחשב על השולחן = what I was asked to do

This is a very common pattern: context + verb + person + infinitives.

You could rearrange parts in other contexts, but this version sounds smooth and standard.

Yes. Even though Hebrew uses מ־ (from) after לבקש in this structure, the natural English translation is simply:

• They asked me to open the bag...

So when learning Hebrew, it helps to remember this as a normal Hebrew pattern:

• לבקש ממישהו לעשות משהו
  = to ask someone to do something

Not everything has to match English word-for-word.

The main verbs are:

• ביקשו ← dictionary form: לבקש = to ask / request
• לפתוח ← dictionary form: לפתוח = to open
• לשים ← dictionary form: לשים = to put

So if you look them up in a dictionary, you would usually search for:

• לבקש
• לפתוח
• לשים