This is actually a more complicated question than you might imagine, and to better tackle it I am going to break it up into pieces. Also, please forgive my laziness but I am going to plug an older answer of mine that outlines many aspects of Ptolemaic society and culture as well as a more rushed but still useful answer on diplomatic relations between the Roman Republic and Ptolemaic Egypt, these two will help fill in the blanks for some issues that I am going to only briefly cover here so that I can go straight to the heart of you question.

Did she wear Greek robes or have the famous cat eye makeup?

Why not both? Royal iconography was very much subject to variation between contexts so that the same ruler might be self styled as Basileus on coinage and inscriptions intended for a predominantly Hellenised audience but on statuary and inscriptions within a specifically Egyptian context, like temple dedications, would have him titled as Pharaoh of a legitimate Egyptian dynasty. Despite this duality, there is very little evidence of mixing and matching between these two traditions, particularly in the first 2 centuries of Ptolemaic rule, with 1st Century BCE art and iconography demonstrating a greater level of outright fusion instead of just equivalency.

For reference, here is a high quality set of silver coins portraying Ptolemy IV with Diadem, robe, and Eagle if Zeus on reverse and there is a similar set of Cypriot coins depicting Cleopatra and a certain Ptolemy(ies?) with Cornucopia on the reverse as a symbol of Isis/Venus. Coins are pretty great when it comes to determining things like hairstyles and clothing although of course they should not be taken on their own. On the other hand, we have statuary and reliefs depicting several of them in traditional Pharaonic contexts, but these are obviously stylised and symbolic depictions. See this bust of Ptolemy II as Pharaoh and Cleopatra and Caesarion in relief at the Dendera Temple complex.

Cleopatra VII's reign is where things get a bit interesting, because it breaks from tradition in some ways and further develops the attempts of previous Ptolemies at integrating the Egyptian and Greek religions, Dynastic cult, and royal ideology. Although the Ptolemids had been deifying deceased members and officiating them into the Dynastic cult since Ptolemy II, and many had previously identified as one or another royally associated deity (generally Osiris/Dionysius or Amun/Zeus for males, and Isis/Aphrodite for females) she seems to have wielded divinity with an unprecedented level of success. Her father Ptolemy XII is generally viewed quite harshly as his reign proved disastrous in many respects but her policies in relation to the Egyptian elite can be seen as a direct continuation of his reign's unorthodox developments. This policy of building bonds with the priesthoods and the aristocracy of Upper Egypt is actually one of the earliest attested aspects of Cleopatra's reign with her patronage and actual presence at the consecration of the Buchis Bull. Long story short, supposedly Buchis was reincarnated as a bull with specific markings and it was a very big deal in the Thebaid, given that the Buchis Bull died and a new one was reincarnated at the very beginning of her reign this was a good foot in the door for a hearts & minds campaign. What this all has to do with Cleopatra's clothing is that her costume was very much designed with clear political and religious purpose. From her depictions in statue and coin as Isis, Aphrodite, Venus or generally some combination of the two (as in this statue depicting her as Isis/Aphrodite in Egyptian style ), to her meeting with Marc Antony in Tarsus where she sailed down the river dressed

as Venus in a picture

to her appearance at the Donations of Alexandria wearing

the robe sacred to Isis

we can look at how these divine associations were used to better associate with the audience at hand. In Asia Minor she took on the role of Venus/Aphrodite who was already familiar to the Hellenised populace and associated with Astarte, Ishtar and Isis, and was viewed as being the divine consort and counterbalance to the Hellenistic Dionysius, Lord of Asia (that Marc Antony had recently associated himself with as the New Dionysius). In Alexandria she donned the familiar robe of Isis but this probably is not a reference to Isis' sheath dress but the sacred linen robe of the Hellenised Isis who is depicted both as a traditional Hellenistic goddess, with the traditional Egyptian Vulture Headdress, and was depicted as other Ptolemaic queens before her in a bust which is sometimes identified as Cleopatra V her mother. This Hellenised version of Isis would probably have been more appealing to an Alexandrian audience. Technically, there is no reason to think that she would never have donned a somewhat more traditional costume in a heavily Egyptian context, particularly when it came to her association as Pharaoh in Upper Egypt.

That said, you asked a pretty specific question about Cleopatra's fashion and I went off on a mini-tangent, the Ptolemids and most Hellads who settled in Egypt maintained their old way of life in regards to clothing and Cleopatra will have been no exception. As far as royal garb went, they favoured simple silver diadems or the pure white linen headband that marked Macedonian rulers as well as being frequently depicted wearing the Greek himation. Cosmetics are a bit trickier, although they were evidently quite popular in Hellenistic and Roman Egypt and along with perfumes were manufactured and exported throughout the Mediterranean but since most of the surviving literature is moralising about the negative effects of cosmetics. I can not really say one way or the other whether she normally wore "cat-eye" eyeliner, but the Egyptian use of kohl as eyeliner also held religious and spiritual significance as well as being linked to social status, and Cleopatra's association with cosmetic use is well established in ancient literature so...maybe?

Did she live in an era with pyramids?

Well...no, not in the sense that they were still being constructed. The Great Pyramids were constructed around 2,500 years before her lifetime and other pyramid structures were no longer being constructed in her lifetime. But then as now, the pyramids remained a source of wonder and mystery as one of Egypt's greatest architectural marvels, so that Diodorus Siculus in the 1st Century BCE would list it in his 7 wonders of the world and both he and the geographer Strabo wrote about in them in a fair amount of detail. The advent of the Hellenistic era in Egypt saw the construction of new wonders such as the Pharos which was the first recorded lighthouse and used an ever-burning fire and system of mirrors to cast a spotlight perhaps 50 miles out to sea, or the Great Library which housed hundreds of thousands of scrolls and was part of the Museion (more akin to a university than modern museum), but many temples and monuments like the Temple of Hathor at Dendera built in traditional Egyptian fashion. It was a different era from that of the Old Kingdom in many respects but there was very much a continuous history and tradition which was passed down from dynasty to dynasty and survived numerous and repeated foreign invasions and the Ptolemids saw the value in harnessing the iconic power of the Egyptian Pharaohs.

Did her relationship help bring peace between two kingdoms?

Yes and at the same time a resounding no, and in any case not the nations you were probably thinking of. Another user brought up that Greece, Macedonia, and the Seleucid Empire had been subjugated by the Roman Republic. Despite their shared heritage however, the Ptolemaic Kingdom was not on the best of terms with either, from the death of Alexander the founding generals of these dynasties fought over the territories that became the successor states and some regions like Cyprus and Coele-Syria were continuously contested.

The Ptolemaic Kingdom supported the Roman Republic and maintained amicable relations throughout its history but Rome's intervention in Antiochus IV's would-be conquest caused a sea change in the relationship and opened up the door for Roman politicians to intervene in the internal politics of the already declining state. By the time we reach Cleopatra's era in the 1st Century BCE Egypt is a client-kingdom of Rome, and the question of whether to annex it outright has already been repeatedly blocked mainly for fear that whoever carried out this deed would become too popular and wealthy as a result.