Un ingegnere controlla la struttura del ponte in giardino.

In Italian the masculine singular indefinite article is un before vowels and most consonants, and uno only before certain consonant clusters (s + consonant, z, gn, ps, pn, x, y). Since ingegnere begins with a vowel (i), you use un.
Examples:
• un libro, un amico, un ingegnere
• uno studente, uno zaino, uno psicologo

Standard Italian grammar requires an article before common nouns (including professions). You cannot say Ingegnere controlla…; you must say un ingegnere, l’ingegnere, questo ingegnere, etc.
Exceptions (no article) happen in headlines, telegram style or some fixed expressions, but in normal speech and writing the article stays.

Here controlla is the 3rd person singular present indicative of controllare, meaning “to check” or “to inspect.” It does not mean “to control” (as in “dominate”) in this context.
Present tense of controllare:
• io controllo
• tu controlli
• lui/lei controlla
• noi controlliamo
• voi controllate
• loro controllano

• struttura is a feminine noun, so it takes the feminine singular article la.
• ponte is masculine, so its definite article is il. The phrase “of the bridge” uses di + il → del.
If ponte were feminine you’d see della, and without any article you’d wrongly see di ponte (but that rarely stands alone).

• in giardino (no article) is an idiomatic locative to mean “in the garden” in a general sense.
• nel giardino = in + il giardino points to a specific garden (“in the garden over there”).
• al giardino = a + il giardino often implies movement “to the garden.”
When you simply state location (“inside the garden”) without emphasizing which one, you drop the article: in giardino.

You get:
Gli ingegneri controllano le strutture dei ponti nei giardini.
Breakdown:
• gli ingegneri – “the engineers” (masc. pl.; gli before a vowel)
• controllano – 3rd person plural of controllare
• le strutture – “the structures” (fem. pl.)
• dei ponti – di + i (“of the bridges”)
• nei giardini – in + i (“in the gardens”)

Italian has patterns but also exceptions:
• Nouns ending in -a are usually feminine (e.g. struttura, casa, porta).
• Nouns ending in -o are usually masculine (e.g. giardino, libro, amico).
• Nouns ending in -e can be either (e.g. ponte is masculine, classe is feminine).
When in doubt, learn from a good dictionary or memorize the article that comes with the noun.

Yes. Italian allows you to move adverbial/prepositional phrases for emphasis. You could say:
In giardino un ingegnere controlla la struttura del ponte.
This simply highlights location before the rest of the sentence.