What type of plastic is best for outdoor usage

You normally want to use outdoor flooded cable (also called gel-filled) as it is designed to weather the elements. I have a few clients who have gone with outdoor flooded & shielded cable as they were worried about interference on the outdoor runs. Also look for UV-resistant cable if it's not going to be buried or otherwise enclosed.

As for Cat 5e versus Cat 6, for home use it probably won't matter as you most likely running 100/1000B-T Ethernet and it will run fine on both Cat 5e or Cat 6. If you want to try to future-proof your cable--or the price difference is negligible--you can go for Cat 6.

The surprising (to me) answer may be ABS cement. You probably won't find this in the hardware store glue isle, instead head over to plumbing:

You'll want the smallest can possible (e.g. 4oz), and after use to seal it in a zipper lock bag and store in a cool dry location. The stuff goes bad in the can pretty fast.

The ABS cement worked for me far better than any of the other glues mentioned above. You can also cut out a small piece of ABS drain pipe, or other ABS item, to use as a patch for the repair.

Note that ABS cement is a mix of a solvent and ABS powder: in the end you really end up welding the plastic more than gluing it. For this reason clamping is recommended.