He was never a man to seek the limelight.

Prosím, aká je to gramatika ? Je tu na to článok ?

Mne by tam automaticky pasovalo :

He was never a man seeking the limelight.

Je to redukovaná vedlejší vztažná věta. Infinitivem obvykle redukujeme věty s THE ONLY, THE FIRST/LAST nebo se superlativem…

The only person to …
The first man to…
The slowest one to…

Zde to sice nespadá ani pod jednu z možností, ale jedná se o stejný jev. Nebyl to člověk, který by hledal… Popisuje jeho vlastnost, něco běžného, něco, co by ho definovalo.

A man seeking – to je prostě „muž, který hledá/hledal…“

She woke from an uneasy sleep to find the house empty.

(Zobudila sa z nepokojného spánku a našla prázdny dom). ?

Prosím, je to ten istý prípad ? Ako by vyzerala plná/pôvodná veta ? Ďakujem veľmi pekne.

She woke from an uneasy sleep to find the house empty is a variation of the only + to-infinitive construction described below, but it omits the word „only“, which is acceptable in your sentence.

Only + to-infinitive 61e

Only is used before a to-infinitive in order to introduce an action or event which is sudden and a little unexpected. This use is usually formal and literary:

• The music shop ordered 50 copies of the CD, only to discover that the songs were sung in a foreign language.
• He stopped the car, only to start it again violently.
• When the Second World War broke out, he tried to enlist, only to be rejected on the grounds of ill-health.

(Cambridge Grammar of English, Ronald Carter, Michael McCarthy, CUP, 2006, 61e, pp122–123)

Your translation is correct.

je to ten istý prípad ? Ako by vyzerala plná/pôvodná veta ?

No, this usage is a completely different kettle of fish [= něco úplně jiného ] from the previous query. The complete sentence would be She woke from an uneasy sleep and found (to her surprise) that the house was empty.

Thank you very much, Dan.

By the way, for your idiom „to be a completely different kettle of fisch“, we have a lovely version too, „byť z úplne iného súdka“. 🙂 Now, I have learnt two nice pieces of information.