Partizip I
Partizip I is active voice and present tense.
The subject of the participle phrase is always given by the outer sentence and cannot be changed inside the phrase itself. You can basically add all the other valences that the verb has to the construction by putting them before the participle, whether it be objects, adverbs, prepositionals, or other parts of speech, even Relativsätze.
Examples:
Objects: Der seinem Bruder einen langen Brief schreibende Mann gähnte.
Adverb: Der sorgfältig einen langen Brief schreibende Mann gähnte.
Prepositional: Der in einem Café sorgfältig einen langen Brief schreibende Mann gähnte.
Partizip als Adverb im Partizip: Der in einem Café den ganzen Tag an einer Tasse Kaffee nippend sorgfältig einen langen Brief schreibende Mann gähnte.
The order of the additional parts of speech in the construction is the same as it would be in a main clause ("Der Mann schreibt [seinem Bruder einen langen Brief].").
Partizip II
As a main participle in participial phrases, Partizip II is passive voice and past tense. So subjects and objects play a different role, like they do in any sentence in passive voice. Otherwise, like with participial phrases with Partizip I, the subject is determined by the outer sentence and cannot be changed in the participial construction, and you can add parts of speech before the participle.
Karl las den spannend geschriebenen Brief.
Karl las den von seinem Freund Tom seinem Bruder Tim, der in Amerika lebte, auf luxuriösem Papier mit Tinte spannend geschriebenen Brief.
Note how Brief is the subject of the participial phrase, but at the same time the object in the outer main phrase. Also, because it's passive voice, the person who wrote the letter is not the subjevt, but appears in the "von" prepositional in the longer example. Also note the dative object "seinem Bruder Tim" that is a valence of the verb "schreiben" (Jemandem etwas schreiben).
Caveat: participles of modal verbs
Participles of more complex predicates including modal verbs are in some cases also possible (somewhat pathological). These help to escape the limitations about active and passive voice as well as present and past tense.
Passive voice, present tense: Ein anderer, gerade rasiert werdender Kunde mischte sich in das Gespräch zwischen Bodo und seiner Friseurin ein.
Active voice, past tense: Nach einem langen Tag, unzählige Kunden frisiert habend, sank die Friseurin erschöpft ins Bett.
(Also note the commas in the second example. These are used to separate parts that belong to the participle frome parts that belong to the main clause.)
Stylistic considerations
It is a question of style at what point it makes more sense to use a relative clause. The respective last examples above are grammatically possible but will be hard to understand and sound artificial and bloated.