It could be construction adhesive, which would be hard to completely remove. Use a sharp utility knife to open up the crack. Don't worry about removing material that is still providing a seal, just remove the rough stuff in the opening so the new caulk will be able to bond to solid material. Fill the enlarged opening with caulk and you're done, unless you feel the need to paint it.
BTW, you can't paint conventional silicone caulk.
This is a pretty standard screen repair. You need to remove the spline to get the screen out, which can be done with an awl or a narrow flat-head screwdriver.
You'll need a new screen, spline, a spline install tool (under $10), and a sharp utility knife.
It's not very dificult, but does take some patience.
It looks like this is a fixed window, so you'd have to install in place.
Position the new screen, and then push the spline into the channel, pulling the screen tight as you go. I'd start with the top edge.
It's critical to get the first edge straight. Start with enough spline sticking out so you can wrap it down both sides, and on one side, also run back along the bottom. You should end up with a single piece of spline with the ends in one corner (though this isn't crtical, and if you have to cut it that's fine).
Work your way down both sides at the same time, keeping it tight as you go, then finally install the bottom edge.
Finally, cut the excess screen off using the knife.
Here's a full guide (and the source of these pictures): http://www.familyhandyman.com/windows/repair/how-to-fix-a-window-screen/view-all
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Just above the sash, in the 2nd pic, can you push that track in (towards the window frame)? If so, follow along, if not skip down to "Continue Here".
If so, you simply push both tracks in and pull the top of the sash out. This will allow you to tip the window horizontally into the room. Use this feature to wash the outside glass.
There will be some sort of mechanism holding the bottom of the sash into the track. Usually this is disengaged by lifting one side of the sash, allowing it to slide up, while holding the other side of the sash in place. This will disengage these lower clips allowing you to completely remove the sash from the opening. Take a look now to ensure you understand how the mechanism latches into place. Maybe even take a couple of pics so you don't forget. Are there moving pieces? If so, how are they arranged immediately upon removal? They'll likely move at some point during your work and you'll need to know how to put them back so you can reattach the sash later. Trust me on this - you'll save yourself frustration later. Don't ask how I know these things... :/
In order to completely remove the track, you'll want to first remove both sashes (top and bottom). Since these look like replacement windows in the original opening, it's possible that the tracks are simply screwed in place and once you've got the sashes out of the way you'll be able to unscrew them. If they don't unscrew then...
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You'll need to remove the stop. That is the piece of wood you see on the inside of the house right next to the track.
Using a sharp (new) utility knife, carefully slice the paint between the stop and the window trim (to prevent the removal process from pulling up extra paint). Then gently work a stiff putty knife or other thin pry bar into that slight gap. If you have to pry, I'd suggest a scrap piece of wood under whatever you're prying with to prevent the metal from digging into the wood and leaving a gouge that you'll later have to patch up.
Once you've got the stops out of the way, you should be able to figure out how to remove the tracks.
I'm not sure if you'll be able to reattach the springs (the one in the pic looks like it's attached and stretched slightly, maybe it's the one on the other side that isn't). If the spring is broken, you'd have to replace it.
If a replacement is necessary... I'm not sure. You don't seem to know who manufactured the windows - maybe there is identifying info on the back of the track. Contact them to see if there's a warranty (our windows, replaced last summer come with a warranty that's transferable to the next owner), or if you can buy replacements parts or whole tracks. You'll probably need to supply them with the dimensions of the sash, as the springs are balanced against the weight of the window.
If there are no manufacturer marks on the tracks (unlikely, I'd think), you'll probably have to contact a window company to see if they can help you out. You may be able to get a "generic" replacement, or you might find someone with the knowledge to identify the manufacturer based on the track and attachment mechanism design.
In any case, it may take anywhere from a day to a few weeks to get replacement parts, so be prepared to replace the sashes while you're waiting. Reassemble in the opposite order of disassembly. If you have to pry the stop off, nail it back in, but leave the nails proud so it's easy to pull them when your replacement parts come in. You don't want to have to go through all that work of prying it off a second time!