So I must update this for everyone suffering from this. I've finally figured everything out:
How to Import SRT caption files into Premiere:
1- Load the video file in your library.
2- Drag video file into timeline
3- Load SRT file in your library
4- Right click video file in library, properties. Take note of resolution and FPS
5- Right click SRT file in library, modify, captions, choose Open Captions. Change timebase to video FPS and Video settings to video resolution.
6- Drag SRT file into timeline.
7- Double click captions object, copy first caption IN time. Paste IN time into timeline's timer.
8- Drag Captions object to the blue line.
9- Edit font size, color, whatever else you want.
Now your subtitles made elsewhere and exported into SRT will play correctly.
How to merge several SRT files into one SRT file:
10- After doing the steps above for each SRT file, SHIFT+CLICK on every SRT file on your library you want to merge into one.
11- Right Click the last one, choose NEW SEQUENCE FROM CLIP
12- A new timeline object with the name of the new sequence will open, no video available. All caption objects will be inside, maybe in disorder.
13- Place them in desired order. Repeat step 7 and 8 above for each to do so.
14- Close this timeline object, go back to library.
Now all your captions are merged into one big caption in your library.
How to export captions:
15- Select SRT file in library or merged captions file
16- Go to FILE»EXPORT»CAPTIONS and choose SRT
17- Select save location.
Now you have a new SRT file for your captions.
IMPORTANT NOTE: If you open your Premiere exported captions file in another software like SubtitleEdit, Aegisub, Notepad, etc. you will probably notice that IN and OUT times don't match. I'm not sure why this happens but it is a visual issue only. If you actually play your video from Aegisub with your exported captions loaded everything will be synchronized. If you play your video and load your captions on VLC player, they will be synced too. I don't know why Premiere shows some IN and OUT times and other software shows different IN and OUT times but in reality everything is synced correctly.
If you notice your captions are not synced, you probably failed at steps 7, 8 or 13.