I've not done Asylum yet (I will be doing it in July), but I will echo the other users: those spin-lock weights will be FRUSTRATING.

I tried to use a set of those with P90X and failed miserably. I couldn't keep up with the weight changes. Pause... spin... spin... spin... spin... "shit I'm cooled down."

I use PowerBlocks. The sport version works really well with P90X, but if you're doing Asylum, you could probably get away with a few random hex weights. With the PowerBlock Sport 5s, I use Plate Mates to improve the weight range; so instead of just being able to do 5 pound increments, I can do 2.5 pound increments. Why didn't I buy the Sport 5.5 (which gives that increment)? I couldn't get it in Canada. :(

I also own the Urethane 90 PowerBlocks too, but those are harder to change (the weight increments take longer to switch to because of the in-handle adder weights). Regardless, the U90s are for later when I do Body Beast (I'm also going to buy the EZ-Curl bar). I really love my PowerBlocks...

Honestly, I don't know what to recommend to you if you're not willing to invest a little in your equipment. It's one thing to try and save some money, but when it comes to weights, you're going to have to spend something. Those spin lock ones are brutal, and I bought a couple sets of them to try and compensate for the terrible weight change times.

After I realized how shitty the spin-lock weights were, I did some research on what to invest in. It came down to BowFlex SelectTechs or PowerBlocks. The SelectTechs are an inferior product; they've recently changed certain components to plastic instead of metal, so they tend to wear down quickly, whereas the PowerBlock build quality is top notch. There are other options, like the Golds Gym adjustable dumbells, but I can't really speak to the quality of those. You could also try something like this but again, I can't speak to the quality.

As for pull-up bars, I don't know where you're planning on installing it. I own 3 different types; a Perfect Pull-Up bar a knock-off bar that looks very much like this one and an actual pull-up sled (which I don't even use). The pull-up station doesn't work well because you can't do close-grip stuff, but the others work fine.

Of note, the Perfect Pull-up can be installed in any door frame, so long as you're willing to drill into it, whereas the other pull-up bar only fits in standard width doorways (32"). My bar is currently in my parent's basement, but there was no where to put it. All the doors are all against walls, so there was no room. What ended up happening with that bar is I built a support system into the floor joists (the basement is unfinished) out of 2x4s and it supports my weight.

TL;DR: So, really, it's up to you. Any bar will work, so long as you know the dimensions of both the bar and where you plan on using it. And any weights will do, just some will work better than others. The spin-locks WILL be frustrating, though.