does this connect with an r6 wire?

Do you mean "RG-6"? RG-6 is a type of coaxial cable. Another widely used coaxial cable is the lower-grade RG-59. Both RG-6 and RG-59 are used for antenna leads, for cable TV, and in some cases for broadband Internet. Both use "F Connectors" on each end. The "extension adaptor" shown on this page will join two F connectors, so it can be used to join two RG-6 cables.

Are these able to work outside without being effected by weather?

I would think not. Most outdoor connectors that I've seen have a rubber sheath over them to protect against the weather. I would look for one specifically made for outdoor use (or find some kind of protector you can cover the connection with I guess.)

I just want to use this to be able to move out TV to another wall so that we can have more room. Will this work well for that?

Absolutely. I used it (with another coax cable) to move both my cable and modem across the room. I lost no quality.

Is this for cable cords too?

No, just the connector that lets you join two cables with f connectors installed on their ends. Everyone's cord length requirements will be different.