For Safari acting weird, try going to Settings > Safari and Clear History and Website Data. Old cookies can sometimes cause all kinds of problems with a site. While you are there, make sure Block Cookies is set to Allow from Current Website Only. This is the most secure yet still useful settings. Block Always breaks too many websites.
I have two things you can try for the Safari/Internet speed problem. The first is to turn off your router, then go to Settings > General > Reset and Reset Network Settings. When the router comes back on line re-join your network.
The other is to go into your router and either turn off the 5 GHz band, or change channels. Another member was having connection problems with his iPad Air, and found that turning off the 5 GHz band stopped the disconnects.
Thinking that this might be caused by interference or the iPad not dealing with the band/channels well, I went into my router and changed the the 5 GHz channel from 153 to automatic (which probably changed the channel being used). Since then my Safari browsing has been much better.
I'm not certain this is a fix. I've had reprieves from the slow internet problem before. Time will tell.
There's not much I can do to help with FaceBook. It hasn't been updated for iOS 8 yet; but make sure you do have the latest update anyway. They say they are doing updates every four weeks, so the next one should be out soonish. If you get Safari working well, you can try using the website for a while and see if you have better luck.
I'm not sure what you are talking about with the pop-up buttons and filing in fields. If you are talking about Safari, then fixing the slow internet should take care of most of it. Remember that when dealing with small text boxes in Safari you can use the two finger pinch in/out gesture to zoom in, then scroll the text box for ideal viewing.
If you pinch in too far you'll get the pages view. Just pinch out again to go back.
If none of this works, you may need to do a restore from a computer using iTunes. There are instructions for that around here somewhere. If the above does not work, and you don't stumble across the link reading other posts, say something and I'll hunt it down.
Edit: Belay that last. Johanna has beaten me to that part while I was tap-typing away.
