Q1. Did Sita take or wear any Jewellery when she departed to the forest? What did she take?
Sita had Jewellery while leaving Ayodhya. Later she got more Jewellery from great female ascetic(also wife of great Sage Atri) Anansuya to Sita. She gave it to Sita when Shree Raama did visit Ashram of Rishi Atri (during vanvas).
idam divyam varam maalyam vastram aabharaNaani ca |
anga raagam ca vaidehi mahaa arham anulepanam || 2-118-18
mayaa dattam idam siite tava gaatraaNi shobhayet |
anuruupam asamkliShTam nityam eva bhaviShyati || 2-118-19
"O, Seetha, the daughter of Videha kingdom! Here are divine gifts: a garland, an apparel, jewels, a scented cosmetic and rare body-cream. These are all given by me to adorn your limbs. They will be ever worthy of your and will remain in tact (even after constant use)."
anga raageNa divyena lipta angii janaka aatmaje |
shobhayiShyaami bhartaaram yathaa shriir viShNum avyayam || 2-118-20
"O, Seetha! Your body, anointed with these heavenly cosmetics, will cause your husband to look beautiful, as Lakshmi (the goddess of fortune and beauty) does the imperishable Vishnu (the Lord of Preservation)."
Q3., Q4 and Q5 explained below
Finding them wasn't big issue because they saw Sita throwing Jewels from sky. Since they seen Seeta Throwing jewels, so they already aware from where to pick that jewels since they seen Jewels falling from their own eyes. Hence there can't be any trouble fetching jewels if they already knows where they fell.
त्यज शोकम् महाबाहो ताम् कान्ताम् आनयामि ते | ४-६-८
अनुमानात् तु जानामि मैथिली सा न संशयः |
ह्रियमाणा मया दृष्टा रक्षसा रौउद्र कर्मणा || ४-६-९
क्रोशन्ती राम रामेति लक्ष्मणेति च विस्वरम् |
स्फुरन्ती रावणस्य अन्के पन्नगेन्द्र वधूः यथा || ४-६-१०
"Give-up your lamentation, oh, great-shouldered one, for I will bring back that lady for you. And by inference I know her... undoubtedly I have seen Maithilee while that demon with cruel deeds was abducting her... then she was crying piteously calling 'Rama, Rama...' and also calling 'Lakshmana...' and she was writhing in the flanks of Ravana like the wife of serpent's lord...
Highlighted line in above quote in which Sugreeva is telling Raama that he has seen Seeta.
आत्मना पञ्चमम् माम् हि दृष्ट्वा शैल तले स्थितम् |
उत्तरीयम् तया त्यक्तम् शुभानि आभरणानि च || ४-६-११
"Myself being the fifth one on the cliff of the mountain, on seeing me she dropped the upper cloth of her sari along with auspicious jewellery. [4-6-11]
Here Sugreev told that they were 5 (including himself) who seen Seeta. And Seeta throw Jewels only after seeing them so that he can take those Jewels and gave to her husband and tell who took her away.
तानि अस्माभिः गृहीतानि निहितानि च राघव |
आनयिष्यामि अहम् तानि प्रत्यभिज्ञातुम् अर्हसि || ४-६-१२
"We took those jewellery items and stored them, Raaghava... I will get them and it will be apt of you to identify them..." thus said Sugreeva to Rama. [4-6-12]
Q2. How many Jewellery items did she drop to guide Rama when Ravana was in the process of abducting her?
शाद्वलिन्याम् ध्रुवम् भूम्याम् सीतया ह्रियमाणया |
उत्सृष्टम् भूषणाम् इदम् तथा रूपम् हि दृश्यते || ४-६-२१
"Seetha must have definitely dropped these ornaments onto pastureland, for they appear to be in their original shape, indeed." So said Rama to Lakshmana. [4-6-21]
एवम् उक्तसः तु रामेण लक्ष्मणो वाक्यम् इदम् अब्रवीत् |
न अहम् जानामि केयूरे न अहम् जानामि कुण्डले || ४-६-२२
नूपुरे तु अभिजनामि नित्यम् पाद अभिवंदनात् |
When Rama thus said Lakshmana said this, "I know not the bracelets and I know not the earrings But because I always salute her feet I know these anklets well...[4-6-22]