You want the silver-top 528MP MaxPerformance at Lowe's or a hardware store (or you can get it online from Lowe's or other suppliers). Lowe's calls it an "MPK" because of their weird numbering nomenclature, but it's the same thing. In fact the package will say MP. You want the silver top one, and there is only one silver-top one, which is the MP (or MPK or whatever the particular store wants to list it as online).
You want that one because most fill valves have a refill ratio of 20% That's the percentage of incoming water that goes through the little hose and down the overflow riser to fill the bowl, rather than out the valve and into the tank. That was an industry standard that in the past just overfilled the bowl and put a lot of water into the bowl on refill that just ran down the drain. Toilets that use less water in the flush, and have shorter refill times for the tank, need a higher refill ratio than 20%. The Toto GMax and Power Gravity toilets run a 40% refill ratio. (The newer models run between like 28% and 33%, hence the adjuster so that valve will work on any Toto.) The 528MP has a little adjuster on it that varies the refill percentage from zero to 40%. So on yours, you probably just need to run it wide open. You don't need the adjuster, but you need the 40%. So a white-cap 528 at 20% (which is what HD carries) isn't going to sufficiently refill your bowl. Make sense?
Now, that said, there is a 528T Korky fill valve that is fixed at a 40% refill ratio. It has a blue top. It is much-less-widely-available than the 528MP, but it is out there in a lot of hardware stores. The reason we don't talk about it much on here is that the 528MP is often available in places that sell it for less than you could find the 528T. ("T" stands for Toto, which was cool when most Totos were Power Gravity or GMax. Now that there are other flushes, the 528T doesn't actually work on all Totos, although it will work on yours.)
The Big Box stores carry a weird assortment of Korky products. The 528 with the white cap is available at HD, but they don't typically carry the silver-cap MP. Lowe's does (yay!), but then Lowe's doesn't carry the r528 cap that I refer to later in this post, and HD does. Lowe's doesn't carry the Blue Fits Toto Korky flapper for your toilet, but HD does online and in some of its stores.
http://www.homedepot.com/p/Korky-3-in-Toilet-Tank-Flapper-for-Toto-Power-Gravity-2022CM/202666005 Oddly, Walmart carries that flapper online and in some of its stores, and carries the 528T online but not in most of its stores. Searching all of those web sites for Korky products is confounded by the crappy search engines they all use, and their tendency to use non-standard letters after the Korky product number. So I just search by putting in Korky, which doesn't pull up more than 15 or 20 things, and then just scroll down to find what I want.
If you have a Grainger around, you can ask them if they have the 528MP and/or the 528PRO (which is a 528MP with a black top and some other tweaks that pros like Terry use). They carry it, but not in all stores.
You can also find the 528T (which comes in a box that says FITS TOTO on it) in many local Ace Hardware's and such. Some Ace Hardware stores also carry the 528MP. Problem is that most of their "helpful hardware folks" act like they know about toilets and plumbing in the kind of detail I am talking about, but they don't. And they will do you wrong in a second. "Oh just get the white-cap one." or "This brand works just as well." Sigh.
Also, if you have a good local plumbing supply place, you could get yourself a Toto TSU99A.X Universal Replacement Fill Valve. It's -- guess what? -- basically a 528MP with some gizmos that let you adjust it perfectly to the factory specs for those Totos that use between 20 and 40 percent refill.
So...those are some good options. Also, if you have a low-cost shipping option, you can order it online. But I suspect that you would rather have it right away.
NOW -- your other questions.
The thing that you broke was likely the little plastic tab that keeps people from tampering with the level to which you set the fill valve. See, to adjust the water level on a 528, you twist it to unlock and then set the length and then twist to lock. To tweak it once you turn the water back on, you turn off the water, twist it to unlock, and then move it slightly up or down, and twist to lock again. When plumbers (and Toto) install that valve, they put that little tamper-proof-key on there to keep it from turning (i.e. from a malicious or clueless person). Korky generally recommends now that you don't really need it in a do it yourself application, but they provide it anyway. I don't bother with it on toilets in my house. It's a PITA to remove unless you take the valve out of the toilet, which is I guess the point to prevent tampering, but it just makes it annoying to adjust. In fact, the OLD installation method recommended by Korky was to dry-fit the valve with the thing initially fully-extended, then push it down to whatever level you thought was correct, then carefully remove it and only then lock it and install the tamper proof key, then replace it in the toilet and tighten it down. That works, but it ignores the concept of wanting to adjust it, and was such a hassle that they just skipped the tamper-proof key in their more recent instructions, and tell you to lock it in place while it's in the toilet. Be sure to hold the stem of the valve (not the head) when you are tightening the nut under the tank because otherwise on the last turn or two of the nut, the whole valve will twist in the toilet and not be facing where you want it.
The thing you would have replaced is this:
http://www.homedepot.com/p/Korky-Toilet-Fill-Valve-Replacement-Cap-R528CM/202666007?keyword=r528 That little cap has all the moving (or wear) parts in the valve, so when the valve finally starts to wear out, all you do is pull off the top, swap out that cap, and you're back in business. We usually recommend that you clean it at the same time, but that takes less than 5 minutes. A nice feature of the Korky valves. That same cap works on every version of the 528; it's not specific to any model of 528. Here's the video on how to install it. If you want, you could try replacing it now, and doing the rest of the service in this video, but it just seemed to me that it would be easier to swap the valve if you're not sure if you broke it.
[video=youtube;syJY4o90prI]https://www.youtube.com/watch?v=syJY4o90prI[/video]
PS One thing about that video is that seems to suggest from what the lady does with her finger that you should be able to spin the rubber within the cap. You can't, and that's not what they are meaning to say; Korky and I had a discussion about this, and they may fix it in future videos. All they mean is that it should have some give to it and move freely up and down. After years at excess water pressure, that cap will have big indentations in the rubber.
AND...to answer your other question, when the toilet has finished the refill cycle, nothing should be coming out of the refill hose into the overflow riser or into the toilet. When the valve turns off, it should stay off until your next flush. If it doesn't stay off until the next flush, and doesn't overflow the top of the overflow riser as yours did, then some water may be leaking past the flapper and the valve is cycling on and off to maintain the level of water in the tank (i.e. flapper is worn out). But if it's coming on and overflowing the top of the overflow riser, then it either isn't adjusted to the proper height (unlikely as it has been in there all this time and doesn't come loose unless someone tampered with it, but you had your key installed) or, most likely, it needed a new little r528 cap.