Apple have no method to connect iOS device to the TC at any time.. LAN or WAN.. so any method you use must come from 3rd party app.
I heard about filebrowser but it doesn't work well.
It works as well as any other app can. Remote access must be based on a network protocol and the only ones supported by Apple are AFP and SMB.
AFAIK people have not been successful getting AFP based file apps on iOS to work remotely although I have not checked closely for a while.
So you have to use SMB.. and at least filebrowser has instructions.
http://www.stratospherix.com/support/gsw_timecapsule.php?page=6remote
It is however unsafe opening SMB at any time. And I suggested people try it with different ports instead of obvious ones.. but that didn't work so why it needs to be 20445 I don't know.. but it does make it fairly obvious.
The alternative method which is really the only safe way to do this is using a different router.. one with built in vpn servers. These used to be expensive but you can buy them now for less than $100.. or use routers with 3rd party. You have to move the TC to being bridged and used as WAP and switch. However that is not a major issue. The vpn router will include a standard ddns client so that fixes not having static IP.
And iOS has good vpn clients built in or app add ons.
VPN is the right way to do this.
And what about apple TV, can i stream from my TC?
TC is a dumb hard disk in a router.. it has no media extensions. Apple TV is not a media player.. it is a streamer. So you must have an application which can stream from TC to the ATV which means you need a computer with iTunes or you need ipad and some software that can read files and stream it.. obviously apple built the system for itunes and itunes pretty much exclusively.
If you get a WD live TV it can play source material directly from TC.. because it is a true media player. Apple TV cannot.