First thing that stands out...
note the long rod on the left with two screws.
It sounds like that is not set to the correct height to trip the shutter as well as the mirror.
If you have already taken apart the camera or started just to remove the bottom for a look around. This is the time to start making notes as you go.
It is always tricky to instruct or advise over the internet and more times than not that a part will be missed.
You should start first with a clear work area and time to make note as well as look at the service manual for some details.
The disassembly process is worth looking at from the manual and make sure some of the basic tools are available.
The responses to you above answers will follow.
Slow speed and mirror cage - as well as the film advance are linked together. The picture taking process starts basically from the mirror going up, but it sounds like the release is out of alignment.
Focusing Screen - beside scratches, is it actually still accurately focusing.
Foam of mirror - missing or is it decayed to a point were it is a sticky strip?
Light seals - they are also originally a foam and should have decayed at about the same time as the mirror foam.
Shutter Curtain - that is good news. Many times I see signs of it being toyed with as the assumption that it was caught on something when a lag occurs.
LEDs - If they light up is good news. How they light up will be fixed when the rod or leaf switch is put right or if the "Av volume" is checked.
X, FP - You'll have plenty of time to check that after the mount face is opened.
Water damage - A common problem and you can get it by spills, rain or snow on the top as well as flooding or camera bag filled with water for the bottom.
The first thing is to strip the camera down.
Top deck: Rewind knob, spanner nut, trim washer on the rewinder side. Two pin screw turned clockwise to remove and trim ring, small two pin screw at center of the shutter speed dial and lift up and don't turn over and the trim ring. The release button need not be unscrewed and just the three deck screws.
Leatherette must be peeled from the front to expose the mount face screws. Timer are need not be removed... care must be taken not to loose the linkage pin under while removing.
More later if you are going to do all this.