How to define $\mathbb{Q}_p(\sqrt{5})$ and $\mathbb{Q}_p (\sqrt{5} ,\sqrt{3})$ and find their valuation rings for $p=7$?

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Try:

sage: Qp=pAdicField(7)
sage: R.<x>=Qp[]
sage: K.<a>=Qp.extension(x^2-5)
sage: OK=K.integer_ring()
sage: OK
7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_7$, so $\mathbb{Q}_7(\sqrt{3},\sqrt{5})$ is just $\mathbb{Q}_7(\sqrt{5})$.