Safe zones have nothing to do with how the video is exported. They're just meant to give you an idea of how some (ancient) television hardware might obscure the edges of the video. This is less and less relevant these days, as digital television hardware doesn't need to use overscan, and most televisions don't have as much of a bezel.
What you're seeing when you only see a portion of your video in the Program Monitor is the result of not having a sequence that matches the dimensions of your footage.
Since you're obviously new to Premiere Pro, I strongly recommend that you work your way through some basic learning materials first. Doing so will prevent a lot of frustration.
Sounds like multiple issues.
1st, if your premiere sequence is set up properly to match your video source, your video should fill the screen entirely, overlapping title\action safe boundaries.
2nd, the title safe area is going to affect your old standard definition 3x4 monitors more than anything. Computer, Youtube, streaming, etc safespace will not come into play.
HD broadcast vary a little, and while there are safespace guidelines for HD Editing, the whole picture tends to be shown on the HD Screen.
Most Consumer HD TV models also have several zoom and aspect ratio modes to adjust for various broadcast signals thrown it it.
3, You can crop your video any way that you want. My choice for doing this is the 4 Point Garbage matte Effect in the Video Effects> Keying bin.
Also, You can resize and reposition your clip by selecting it on the timeline, then going to the Video Effects Control tab in the Source Monitor and making adjustments with the Scale & Position values.
Thanks, Stephen_Spider, you were so helpful. It was taking me forever to find the ability to scale and position again. It's especially frustrating if you've done it before and forgot how. It's a pain looking tutorials over when they don't show you what you want.