The Bible, like any other Holy Book, is to be taken as a whole. When you get past the Pentateuch, try again. For Christians, that includes the New Testament. So try to include that with your arguments.
I'll give a few examples to start with.

>
>I want to thank you for your efforts to educate us regarding
>the Bible and God's law. I can learn a great deal from
>you, and I try to share that knowledge with as many
>people as I can.

A little learning is a dangerous thing;
Drink deep, or taste not the Pierian spring:
There shallow draughts intoxicate the brain,
And drinking largely sobers us again. -- Alexander Pope

To paraphrase CS Lewis - Let us discuss books written for adults like adults and not like children.

>When someone tries to defend the
>homosexual lifestyle, for example, I simply remind
>him that Leviticus 18:22 clearly states it to be an
>abomination. End of debate.

We are not to pass judgement, nor is it up to us to forgive sins that were not committed against us personally, which includes a homosexual lifestyle. see I Corinthians 6:18.

>When I burn a bull on the altar as a sacrifice, I
>know it creates a pleasing odor for the Lord (Lev.
>1:9). The problem is my neighbors. They claim the odor
>is not pleasing to them. How should I deal with this?

See Mark 12:33

>
>I would like to sell my daughter into slavery, as
>it suggests in Exodus 21:7. In this day and age, what
>do you think would be a fair price for her?

See I Timothy 1:9-11

>
>I know that I am allowed no contact with a woman
>while she is in her period of menstrual uncleanliness
>(Lev. 15:19-24). The problem is, how do I tell? I have
>tried asking, but most women take offense.

Ibid. btw: that would also be called Adultery - if the woman is your wife, you can always apologize for asking.
Read Lev. 15:1-18 - before worrying about a woman making you unclean, have some consideration for her in not making her unclean.

>
>Lev. 25:44 states that I may buy slaves from the
>nations that are around us. A friend of mine claims
>that this applies to Mexicans but not Canadians. Can
>you clarify?

Only if you're an Israelite living with Moses at the time.

>
>I have a neighbor who insists on working on the
>Sabbath. Exodus 35:2 clearly states he should be put
>to death. Am I morally obligated to kill him myself?
>

Numbers 35:30

>A friend of mine feels that even though eating
>shellfish is an abomination (Lev. 10:10), it is a
>lesser abomination than homosexuality. I don't agree.
>Can you settle this?

James 2:10 - apparently there's no difference; to the Lord sin is sin.
As for food, see Romans 14:20

>
>Lev. 20:20 states that I may not approach the altar of
>God if I have a defect in my sight. I have to admit
>that I wear prescription glasses. Does my vision have
>to be 20/20, or is there some wiggle room here?

Lev. 21:20 (for the others following)
You couldn't approach the altar (according to Leviticus) unless you are a descendant of Aaron, anyway.
The other defects listed therein appear to be more physically apparent, so if you are a descendant of Aaron, you may be alright.

>
>I expect you have studied these things extensively,
>so I am confident you can help. Thank you again for
>reminding us that God's word is eternal and
>unchanging.