Suppose I create an air bags website that many others will link to. 
Would bagsair.com be better than air-bags.com?   Would search engine
software read the link as "keyword: air except bags?  .. Also, when
search keywords are used as domain name is "bag" the same as "bags"?

I'm not a researcher, so my comment is free, of course.

I don't think your domain name has anything to do with it. Search
engines work on meta tags and other items. I personally think bagsair
would be a mistake, and air-bags.com is pretty darn good if you can
get it. Now, hurry up and buy it before someone else does and then
tries to sell it to you. Buy it at www.ev1servers.net (very cheap!). I
swear that they keep track of names that people search availability
on, and then they buy them up based on that, and then jack the price
up. Anyway, go to any *normal* website, like my own: 
www.vbaexpress.com, and from Internet Explorer, hit View-Source.
You'll see this in there:

META NAME="keywords" CONTENT="microsoft,office,microsoftoffice,help,support,
assistance,trouble,troubleshooting,troubleshoot,consultancy,consultant,Add-ins,
spreadsheets,worksheets,workbooks,documents,database,db,data,base,VBA,macros,
utomation,automate,automatic,automatically,access,excel,powerpoint,power,point,
visio,outlook,word,winword,win,windows,free,forum,presentation,slideshow,slide,
show,template,dot,doc,xls,xla,xlt,mdb,mde,ppt,exe,frontpage,front,page,global,
samples,officevba,wordvba,excelvba,accessvba,powerpointvba,pointvba,visiovba,
frontpagevba,outlookvba,cleanyourpc,clean,your,headquarters,experts,theofficeexperts,
theofficeexpertsforum,newsletter,newsletter,letter,samplevba,downloads,myaddiesbook,
mypicturebook,xlfunc,myvlookup,myinvoicing,cutequiz,mytexttools,
excelthesaurus,myformspeller,mystationary,technicalsupport,technical,source
code library"

And the whole purpose of those is to be found by search engines.
Someone mentioned that they search the first 250 words on your home
page. I do believe they mean *behind the scenes* and not necessarily
*out front*.

I hope this makes sense, and I wish you the best of luck!

Actually, it'll see it for both bag and bags, since it contains both.
You can prove this for yourself with a few searches, you'll notice how
Google highlights sections of the domain name which match keywords in
your search.

Register a domain name that looks like something a visitor would want
to click on. There are, unfortunately or not, no real tricks you can
pull these days with domain names. Some people used to swear by
keyword-laden domain names, but nowadays they're not of much added
benefit.

What you need to pay very careful attention to, however, is how people
link to your site, but that's beyond the scope of this question.