Native speakers simply wouldn't normally say "I'll show you into to your room". The two credible possibilities are...
1: "I'll show you to your room"
Appropriate in contexts where the offer is to help you find your way to the room, or simply to act as a "protective escort". The speaker might reasonably leave as soon as you're within sight of the door.
2: "I'll show you your room"
Appropriate in contexts where the offer is to accompany you into the room, perhaps to point out anything that might not be obvious (where the light switches are, how to operate the minibar if it's a hotel room, etc.)
In practice, both contexts often apply simultaneously, and either of the above phrasings could be both intended and understood as implying the other.
For reasons that aren't immediately obvious to me, although I've said into is unlikely in OP's exact context (with a room), it seems perfectly normal to me to...
3: "Show him into the garden"
...which carries no particular implications of presenting the garden - just going with him to get there.
EDIT: In informal contexts such as a guest staying overnight at your house, #2 above (no preposition) is the normal form. In formal contexts (hotel staff, a wealthy person's house-servants, etc.) to is more common, carrying either/both implications of guiding and/or escorting.
The relative uncommon into (almost always formal) implies the speaker will accompany you through the entrance (door, gate, etc.) to wherever you're going, and assist you in "settling in". Thus, anyone showing you into the garden would usually accompany you into the garden and introduce you to your host and/or other guests. If a hotel manager shows you to the dining room, he might well just leave you at the doorway - but if he shows you into the room, he'll probably usher you to a table and see that you're seated before leaving you in the care of the restaurant staff.
Is "assist in" the same as "assist with"? Can I always substitute the one where the other is used, or is there a difference in meaning?
In general, they're pretty similar and frequently interchangeable -- at least one-way. (You can say "assist with" for "assist in," but you should be careful swapping "assist in" for "assist with," since there are some nuances.
If you "assist in," it may indicate a longer-term and/or more in-depth assistance -- the senior academic meeting with the junior one daily over a course of months, for instance.
If you "assist with" something, it can indicate a much shorter-term, or "shallower" form of assistance -- you aren't going around asking questions to help the police, but are instead answering the questions the police gave. (If you said "I am assisting in the police inquiries," you would be presumed to be doing a share of the work.)
So you can assist your co-worker with a project, or you can assist your co-worker in his work, and it means pretty much the same thing. (Though even there, notice the nuance: a project is a thing with an end, while "his work" is more open-ended!) But you want to be careful saying you're assisting the police in their investigations unless you mean that you're doing more than just answering a few questions.
However, even if you use the "wrong" in/with, a sentence in context will usually be interpreted the correct way.
Best Answer
In general, "hunting" implies a high chance of failing to find, catch, or kill whatever/whomever is being hunted.
"Hunt down" implies persevering until whatever/whomever is being hunted is "caught" or "dead" (either literally or figuratively). Often, this involves techniques to "corner" the target(s). A target that has been "backed into a corner" can only escape by going toward the hunter(s).
"Hunt down" is also more likely to imply that particular targets have been identified. For example, "hunt down men" might mean that: