Help exporting small pixel count videos that don't look terrible.

Here are the problems that are difficult to overcome when you are creating small animated GIFs.

1. There are only 256 possible colors available, not 256 for each channel, 256 total different colors.
2. Horizontal and vertical edge detail, like the edges of a font, must be perfectly lined up with the pixel grid to maintain sharp edges.
3. Antialiasing, the trick that makes details look sharp in images with more colors available, like the more than 16 million available in a JPEG, is so limited in a GIF because of the small numbers of available colors that you have to be very careful with the total colors you choose to add to your gif.
4. After Effects does not have a 256 color display option, so there will always be a difference between the colors displayed in the comp panel and the colors you get in your gifs. You can simulate it a little with Posterize, but it will never be 100% accurate.

After Effects also does not let you adjust the timing of each individual frame. It is a pretty poor tool for efficiently creating animated gifs because the text and the color palette are almost impossible to preview in AE.

You can kind of solve the text problem by preparing the artwork in Illustrator, converting text to outlines, then turning on Pixel Preview and Snap to pixel and adjusting your text so that it lines up with the pixel grid. You can also design with only 256 total colors in Illustrator. You could then import your layered AI file as a composition, retaining layer size and work the animation backwards because all of your artwork is in the Hero position before you start.

If you must use After Effects, set the comp frame rate to no more than 15 fps. 12 fps was good enough for Buggs Bunny and just about every cartoon that was every animated by hand, so it is good enough for an animated GIF. Make sure there are no duplicate (identical) frames next to each other (no static images lasting several frames), then export the GIF, open it in Photoshop and set the duration of every frame you want to hold for a few seconds, and do your final export with the Save For Web Legacy option.

An easier solution is to use a software package that is designed to create animated gifs. There are several and they are not that expensive. If you have a full Creative Cloud account Adobe Animate is a much better tool for creating animated GIFs than After Effects.

Most modern, responsive web designers avoid GIF's and use SVG for their animated elements. Adobe Animate is also a great tool for that.

Start by adjusting the kerning of the text so the letters don't visualyl melt together. The rest already has been explained. 350 x 50 is nothing these days and will look tiny even on mobile screens. You can't expect to cram in massive amounts of info in something that will end up as a thumbnail - literally. The whole approach strikes me as slightly questionable, given that most banners are around 600 x 150 at least today.

Mylenium

You can't expect a gif in 350x50 to look high quality. 350x50 is a resolution, and in reality it would be played normally in a very small preview, because it's so low quality. That said, QuickTime or any other video player will "upscale" it, in essence try to make it fit the screen size. That's not a lossless feature, since it doesn't involve any real rendering, and it will lose you a ton of quality.

The only work around to this is (if your client allows you) to preconpose all your layers, resize the comp to something bigger, like 2100x300, and scale the hell out of it your precomp. Then to make it slightly sharper, go under quality & sampling and change it to Draft. Or as a better alternative, if you can, just make everything higher quality. 

https://youtu.be/DuEFByTEevc (he says Draft is bad, but you might want try it out. You can also try Bicubic.)

Theres a high chance what I suggested will make it look blocky. In that case you might be out of luck or someone else might have the answer here.