Chapter 3 of Kalki Purana (unabridged version) has this sloka. Here is complete sloka (from verse 14 to verse 20) in Devanagari and IAST.
Sanskrit verses taken from here. You can also read complete Kalki Purana in Sanskrit and English from here (translated by Bhumipati Das).
कल्कि रुवाच । kalki ruvāca ।
गौरीनाथ विश्वनाथ शरण्यं भूतावासं वासुकी...
How lord Shiva would know that this curse and/or boon is appropriate?
If someone prays to God, then God knows what that person wants:
As per chapter 4 of Kalki Purana:
Understanding that Kamala, whose face is exquisitely beautiful and who
is very dear to Lord Hari, had incarnated within this world, Sankara,
along with Parvarti, joyfully came to see ...
Sanskrit word used is "Yuga" and translation you quoted is given by by Bhumipatidas. Gaudiya Sampradaya (ISKCON) Swamis and Acharyas translate "Yuga" as millenium.
यो धर्मो ब्राह्मणाना हि सा भक्तिर्मम पुप्फला । तयाहं तोषितः
श्रीशः सम्भवामि युगे युगे ॥ १५
The dharma of the brahmanas is to render pure devotional service unto
Me .Being pleased ...
This seems a translation issue, here is the Sanskrit shloka and Hindi translation of Kalki Purana 4.15:
The religion of the Brahmanas is to render pure devotional service
unto Me. Being pleased by their devotional service, I incarnate, along
with Goddess Laksmi (Shri), from age to age.
Are there 10 avatars of Vishnu per kalpa?
There are infinite avatars of ...