The first pair is easily understood if you take notice where さえ attaches to.
When you don't know what piece you should capture, I can only say ビショップさえ取ればいいよ; if you don't know what action you should do to the Bishop (e.g. if you take it, your piece would be eaten too, so which is more favorable?), I can only say ビショップを取りさえすればいいよ.
If your question is only "What should I do in this position?", then both will do.
I also saw on a site that でさえ is the same as さえ in a sentence like 子供さえわかる.
Really? They're very different. Since さえ is a 取り立て助詞 "focus particle(?)" that isn't a primary particle, it generally can only come after another case particle (格助詞). What you see as if さえ comes right after a noun is actually either がさえ or をさえ, but the two (and only they) are usually eaten by 取り立て助詞.
子供さえわかる ← 子供がわかる "understand children" (or could be "children understand")
子供にさえわかる ← 子供にわかる "children can understand" (lit. "be understandable to children")
So, the stumbling block here is, in fact, で.
子供でさえわかる ← 子供でわかる "understand within children / as much as a child can"
This is a case particle, and not a paradigm of copula だ. It's rarely used alone in this sense, though it does exist:
子供でわからなければ、大人に聞くといい。
But instead it typically appears with 取り立て助詞, such as では, でも, ですら or でさえ.
日本語さえ話せない and 日本語も話せない can describe the same situation in most of times, but there is a clear distinction; も comes from "also" thus means it's a deviation from criterion, while さえ implies that itself is a criterion. That is:
日本語（○さえ／×も）話せなければ、聞かれても問題ない。
It doesn't matter if we're heard, as long as (=if only it's met that) they don't speak Japanese.
