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In these two sentences, I would've thought に would be used in place, but
instead を is used? When is it appropriate to use either?

A: 反対側【はんたいがわ】50メートルあたり **を** 探【さが】して！ Search around 50 metres on the opposite
side!

B: もし会【あ】ったらグレース **を** どうするつもりですか？ What are you planning to do **to** Grace if
we see her?

Shouldn't **に** be used in both of these to indicate the direction?
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particles particle-に particle-を
When can を be replaced with に?
I read that に is used for indicating places of existence, target recipient [of
something], the occasion, or the source of receiving an item. (There may be
some I have missed.) I also read that を is for objects of a transitive verb
(meaning a verb that needs something to receive the action it indicates). If I
remember or know correctly, の is used before a prepositional noun like
横｛よこ｝(side)、近｛ちか｝く(near)、東｛ひがし｝(east)、南｛みなみ｝(south)、北｛きた｝(north)、or
西｛にし｝(west) in examples such as フィリピンの南｛みなみ｝(South of the
Philippines)、日本｛にほん｝の北(North of Japan)、電車｛でんしゃ｝の近く(Near the train) 、 etcetera.
I hope this helps.
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Well, generally every language has some quirks in its grammar, but as for your
examples, を is the only option and に is never used. I sense from your
statement "I would've thought に would be used in place, but instead を is used"
that you may think or be taught that case particles each represent some kind
of "universal" trait, but they are actually decided by loose conventional
rules much like English choice of prepositions, that you can't make nearly
perfect prediction from its meaning.

> A: 反対側50メートルあたり **を** 探して！

探す indeed has two kinds of noun alignment: `[target]を[place]で探す` and
`[place]を探す` but neither can take に. You wouldn't see it except as a part of
adverb: 効率的 **に** 探す.

> B: もし会ったらグレース **を** どうするつもりですか？

Don't be fooled by the English translation. The answer you expect for "What do
you do to Grace?" would usually not be "I _do [noun] to_ Grace." but "I
_[verb]_ Grace.", isn't it? In that case, you use the phrase "do what" to
asking another verb; it is a [pro-verb](https://www.merriam-webster.com/words-
at-play/what-is-a-pro-verb-definition-how-to-use), more specifically,
[interrogative
verb](http://phonetics.linguistics.ucla.edu/wpl/issues/wpl17/papers/32_munro.pdf).

English speakers have to work around the lack of pure interrogative verb by
forming an expression "do what" (i.e. "what" as dummy object) and push the
original object out as indirect "to" argument. Japanese doesn't have one
either, but fortunately we have どうする (どう is an adverb) that doesn't take up
the object slot, so that the true object remains the object.

> グレース **を** どうするつもりですか？ vs _What are you going to do **to Grace**?_
>
> グレース **を** 家に帰します vs _I will send **Grace** back home._
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