Whenever I get into the mechanics of cassette playback and you get confused as to what goes where, you can click on the image directly below this paragraph to hopefully clarify things a bit.
tape head misalignment
playback progressively faster
pinch roller issues
Tape decks (as well as personal players, recorders, et cetera) are usually driven by neoprene rubber belts, and they degrade. It's a fact of their existence - they last a few decades and then soften, slacken and finally turn to goop. You can replace them by searching for the specific belts for your device online - eBay is a good place.
If you cannot find a set of belts for your device, you'll have to look for generic belts of the right specifications. For this, you must take three things into account: the length (the total circumference, that is to say the length if the belt were cut in one place and then laid along a ruler), the thickness, and the cross section (usually square, round or flat). It is important to get this right, though there are some tolerances; a very slightly tighter, looser or thinner belt isn't the end of the world.
I'm sorry to say that these loose belts tend to be a bit pricy. Whereas you can expect to pay about fifteen to thirty US dollars for a full set for your deck, you'll likely shell out about twice that if you have to hunt the belts down individually.
Start by clearing out the old belts, and where necessary, using isopropyl alcohol to get rid of any residue. Putting new belts in can be fiddly, sometimes to the point of having to disassemble half the machine. There's no way around it, though. Try to find service manuals or tutorials on how to do it for your device. HiFi Engine is a good place to start looking (you can see if they have manuals for your device, but you'll need to register a free account if you want to download any manuals).
Top tip: if you're looking for belts for smaller devices like personal stereos or memo recorders you could do worse than to buy an assorted handful of belts, also easily found on eBay. These sets will contain various sizes, shapes and thicknesses, and there'll usually be one or two in there that fit your device. These sets are very cheap and handy to have lying around.
Here's one such assorted bag laid out (minus two that I've already used in personal stereo devices), with an AA battery for scale. Your results may vary - not every bag will contain the same sizes. Also keep in mind that you get what you pay for (1 US dollar for this bag), and that not all belts will be perfectly cut.
Another top tip: it is possible to rejuvenate a belt to some degree. Put it in a bowl of water, boil the water for about 10 seconds in a microwave, and the belt will often come out a bit less stretched out and a bit rounder. It's not a permanent fix, but it can do the trick in a pinch.
Get out your contact spray (the kind that leaves no residue) and, whilst depressing the play and record buttons now and then, liberally spray the insides of the switches, and then wait for a while to let all of your contact solution evaporate. If you're in luck, this will restore contact and your deck will play properly again.
If not, here's a simple test to see whether the issue is somewhere in the circuitry or a matter of the tape head: find the wires that come straight from the head and desolder them from the board where they attach. Switch them (the left wire soldered onto the right terminal, the right wire onto the left.) If the sound issue remains the same, there's a problem with the circuitry. If the imbalance is now switched left-to-right, it's the tape head. Don't forget to switch the wires back!
You may or may not be able to fix this. In older and more high quality devices, which were built to be serviced, you'll usually be able to take off the cassette bay door to access a screw that adjusts the positioning of the head:
The tape head is pointing up in this picture. The red arrow indicates the screw that is turned to adjust the head alignment. The yellow arrow indicates the erase head, which can also be adjusted if necessary with the two screws on either side.
Sit in front of the device with your headphones on, play a (properly balanced) tape and adjust the head until the audio is just right.
Top tip: afterward, take a picture of the screw. The balance has shifted before, and it may again slowly slide, so this way you'll be able to see if it's caused by the screw turning. I've not had any such trouble, however.
You'll need to look up a service manual of your deck (HiFiEngine) and find out which potentiometers you can adjust to make one channel louder. The image below is from the service manual for my Akai deck (click for the full schematic); to repair an imbalance, I would turn the potentiometers marked 'PB LEVEL ADJ' (playback level adjust).
Top tip: Take pictures or notes of the positions that the potentiometers are in beforehand. This way, if you can't fix your problem or you screw up, you'll at least be able to restore them to their original positions.
As the tape rolls between the pinch roller and capstan, the uptake spool (which is where the tape is going) exerts some force so that as the tape is fed through the rollers, it takes up the slack and winds the tape onto the uptake reel.
Playback will be too fast if your pinch roller has gone slippery, or if it or the capstan don't slide into place properly. The tape will then be moved along purely by the tension that is put on the uptake spool, which causes the tape to run too quickly.
First of all, check if your capstan is rotating. If it isn't, look for mechanical problems, such as slack belts that need to be replaced.
If the capstan is rotating, check if it and the capstan are making proper contact with the tape.
If they are making contact but the tape is sliding through too quickly, you can rub the roller's rubbery surface with some printer paper to roughen it up.
If the roller isn't moving into place, check the internal mechanisms for snags or broken bits or dried up lubricant - it may be a variety of issues, so there's not much I can help you with there.
An old trick: back in the day, for a laugh, I would cover up the rectangular openings in the cassette through which the pinch roller grabbed the tape, thus creating this no-grip situation intentionally. You can cover up the holes with some plastic film, secured with sticky tape - and make sure nothing sticks to the tape itself. Of course, nowadays anyone can speed up a digital file, but way back when, it was a pretty nifty thing.
Fixing a cracked, gooey, sticky or otherwise broken pinch roller is no mean feat. First things first: if your pinch roller is still in working order, measure it. Find out exactly how thick and long it is so that if it should ever crap out, you'll know what to look for. Minor differences are okay, but if the diameter of a replacement pinch roller is too far off, it may press the tape too hard, resulting in slow playback, or not hard enough, resulting in playback that is too fast.
A sticky pinch roller is often an omen of what is to come, but the stickiness can usually be wiped off with a bit of cloth or some printer paper (this also works for slippery rollers). This will technically make the roller a bit skinnier, reducing its effectiveness, but we're talking fractions of millimetres so don't be too worried.
A gooey or cracked pinch roller needs to be replaced. You could conceivably build one out of electrical tape or restore the surface by sanding it down and adding a layer of pinch roller restorer (very difficult to get flat and even, which is important).
But your best chance of good operation is getting a new pinch roller. Pinch rollers, alas, do not come in standard sizes, so you'll most likely have to look for one that is made specially for your device... it's hard, I'm sorry. If you know exactly the size you need, you'll be able to do better, but still, you'll cry yourself to sleep. If you're lucky, you may be able to find the same device being sold for parts (i.e. broken), so you can source a roller from there.
We need to reapply the lubricant to the tape guide wheels and the parts of the shell that the reels rest on. To get to these spots, you'll have to take your cassette apart. This will usually be a matter of taking out four screws; if there's no screws, look at the 'tapes that won't open up' header down below.
After opening the shell, remove the tape guides, which sit on little spindles inside the cassette (indicated above with circles), from the cassette housing. Spray some silicone lubricant (such as WD40) onto the head of a cotton bud and use it to rub the lubricant on the spindles - it doesn't take much, and you can absorb any drippage with the other end of the cotton bud. Next, remove the tape reels (carefully!) and rub lubricant on the rims inside the cassette housing (indicated above with arrows). Try not to get any lubricant on the tape itself.
After doing all this, replace the reels and guides, put the cassette back together, and you're good to go!
They can wear out or deform, frustrating the smooth rolling of the tape reels. They're hard to repair as such, but they're easily replaced. Get a sacrificial cassette, take it apart, and replace the defective slip mats with the new ones from the donor.
Of course, if you can break the shell apart in such a way that it'll glue back together again, more power to you. I once managed to use a scalpel to weaken the joint and snap the halves apart fairly cleanly, but you shouldn't count on getting it right every time.