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Will signing up for Google Adwords help my site get indexed by Google and other major search engines?
The index is not affected by the ads, but of course ads bring traffic and that may bring links, and links will both get you indexed and improve your position.
If the site is not yet being spidered, just link to it from a site that is and the search engines will find it.
Well, I have heard many rumours that signing up for AdSENSE may speed up indexing. While Google officially denies this, I would be very surprised if MediaBot (the spider visiting AdSense sites) and GoogleBot would NOT share information. Not that GoogleBot rushes at the site once MediaBot was there. But a few weeks later nobody would spot the connection...
But signing up for AdWORDS to speed up indexing sounds a bit far fetched to me. You sure you did not mistake the two programs?
What I usually do with new sites is that I enter them into Googles submission form. Usually GoogleBot comes visiting about 2 days later and usually within a week the site is indexed.