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Are there any algorithms out there that can take a movie, and produce a sharp photo

Of course. And it's free:
RegiStax
http://www.astronomie.be/registax/index.html

It's mostly used by astronomers, but works quite well for any series of images. Of course, it works on repeating frames. If things are moving, them they aren't 'repeating' and won't be processed correctly.

And does it run Linux?

Yes.
RegiStax6 can be run under Linux via wine version 1.3.17
http://www.astronomie.be/registax/linux.html

Unfortunately, the bad news is that you Linux folks will have to write a custom mouse driver with 6802 lines of code to click on the link above.

> Stacking. You can do this at home with a little scope and a CCD. Obviously this is an art requiring extensive signal processing expertise.

I'm pretty sure Photoshop will stack images for you mostly automatically.

RegiStax.
Free.
http://www.astronomie.be/registax/

I'm not trying to troll or be mean spirited, but take it for me, any EQ mount that is in the price range you are looking for is going to drive you absolutely bonkers...at least in the 'motorized' category. You can get close to the upper price range on a manual EQ but it would only be good enough to piggyback a camera. I have been in the hobby (astrophotography) for about 10 years now and have lived through the pain of the cheaper mounts. Unfortunally, cheap mounts are only good for one thing...making you so fed up with them that you eventually get tired of trying and give up....and keep you from actually watching the shower since you will be constantly messing with the mount.

For what you seem to what to do from the summary above you could accomplish with the following :

1. Stick the camera on a normal tripod and aim where the meteors are originating from with the widest angle lens you have.
2. Take 10 sec shots after ten second shots with an ISO of at least 800, while adjusting for position every 10 or so minutes so that you keep the general area of the sky in view.
3. The next day, use an astrophoto stacking program RegiStax to 'stitch' together your images made the previous night.

You would be surprised how good the pictures will actually come out...and for the price of a $50 tripod.

Now, barring that you are dead set on an EQ mount for this, I have only one piece of advice. In the land of EQ mounts, the heavier the better (less shaking, better stepping motors). And the heavier it is, the more expensive it is. There is just no getting around it. I currently have this mount and it required a complete tear down, polishing, and rebuild before it was even capable of astrophotography...and that's at $600!!

..as mentioned above - using image stacking software can make the 'sum' of the images better than any of the component images involved.

There is a free stacking program here:

RegiStax

It works pretty well.

To get into full blown astrophotography you can break down the cost as: 1/3 for the mount 1/3 for the scope and 1/3 for the camera. I tried to go cheap my first round, and it was an exercise in frustration. Currently my set up is around $10,000, and I'm still learning.

But for what it sounds like, go simple. If you are really committed to it, avoid the GOTO scopes - you will thank yourself later. Get as much aperture as you can afford, get a simple mount (a dobsonion setup is a good start once you are determined to spend some money). Get familiar with the skis (this is the key reason for not using a GOTO scope). Find out what you like to view. Is it planets?, Deep Sky Objects (DSO's); galaxies, nebula's, star clusters? Once you've determined that and if you are still interested in doing astrophotography then you can tune the type of telescope to the type of objects you want to image.

If/when you decide to get into astrophotograph, if you have a dark enough sky and the property, build a little observatory (a rolloff is fairly simple). Get a good equatorial mount and build up from there.

IC405 I took this image from the Pawnee Grasslands about 100 miles north of Denver, CO. I used a Takahashi TOA 130 telescope (a 5" refractor) with focal reducer (FL 860mm f/5.73), on a Losmandy G11 mount, ST-4 autoguider, and a Canon 20Da digital SLR. It is two 10 minute exposures at ISO 400, aligned, stacked, and processed using GIMP.

While we're listing astronomy software: Kstars is great too.

And if you want to use a (web)cam on your telescope, take a look at registax.

...the latest version of 'registax' - it's free (beer or speech - i can't remember)....

As in beer, FYI.

What the heck, here's the link, while I'm at it.

TouCam is dirt cheap and perfect for entry level astrophotography. Just google around, there's lots of literature on modifying/using the camera. I myself have taken some nice pictures of jupiter and moon(with filter) using a Toucam and 114mm maksutov-newtonian telescope. After you capture a video w/ the camera you can boot up registax to process it and make a compilation of multiple video frames for a nice still image. If you want to go for imaging deep objects like M31 or other galaxies, you will probably need to invest in a "real" astrophotography CCD.

One of the many sites about the TouCam
Registax

The amateur stacking program Registax seems to be more sophisticated than this. It allows multiple alignment stars or points and shifts the corresponding subregions of the image as needed. Otherwise, the method for selecting the images is very similar.

http://www.astronomie.be/registax/html/multi_opera tion_1_.html

Use something like Registax.