I wondered if everybody was answering the wrong questions.
> how many pins do I need
The common dual-opamps are 8-pin DIP packages, so you need an 8-DIP socket.
> solder tail vs. wire wrap
Wire-wrap has LOONG pins. You use a wrap-tool or machine to connect insulated wires, without stripping or soldering, by wrapping the wire so tight the corners of the pins bite through the insulation. Standard pin length allows three wires to be wrapped on one pin. This makes total sense in LOGIC circuits, which have few or no non-chip components, a LOT of interconnections in a small space, and all wires are driven with powerful outputs that overwhelm crosstalk. In analog work, all these good ideas become bad ideas.
You want SOLDER-TAIL. The legs are just long enough to get through a PCB and clench the bottom.
Gold, tin, open, closed..... buy by price and what you are doing.
If you know what you are doing and do not expect to change chips, don't use a socket! Solder heat will not fry a chip. Sockets are prone to failure, as others have said.
For a project where I only thought I knew what I was doing, and might change chips, I use the cheap tin low-profile solder-tails. About $0.19. Examine them before use: 99.9% of the contacts will be good, but Murphy's Law says if you don't check, you will get a bad contact.
If I was building for posterity, with possible new/improved chips every decade, I might use gold screw-machine sockets. They are more sturdy and pretty. Then again, I might use the cheap tin sockets.
If I was sorting hundreds of chips in a test rig, I might get a ZIF socket. Pin-type CPUs often use these: flip a lever and the chip drops in, flip the lever and the pins are clamped. If you remember seating 386 CPUs with both thumbs and a lot of mobo creaking, or even a 40-pin 286, you know why you need this for many-pin chips. Insertion wears the standard force-fit sockets, so 10 or 100 insertions and the socket gets flaky. Zero Insertion Force makes it possible to put in a 400-pin chip, or makes it possible to insert a hundred chips in the same socket without the fingers (yours or the socket's) getting worn out.
> upgrade a friend's ts-9. I have an RC4559, which I was wondering if that would work otherwise I'll either order a 4558 or pop in a TL072.
I dunno "ts-9", but I find it hard to think a 4558 is an upgrade for anything. And I like the 4558.
4559 is an improved 4559; whether the improvements make any difference depends on the use.
TL072 is a very different way to get about the same thing as a 4558, faster and cheaper. Better for some things, worse for others.