the situation is, I can in no way afford a drum kit, let alone get away with playing one without having the neighbours dig out their pitchforks and knock down my door, so I'm trying to learn a few things on an old, used compact electronic kit every now and then. It's broken in all the wrong places, but it gets the job done, more or less.

So, the thing is, today, I noticed that after ten(!) minutes of playing, I'd managed to get blisters on two fingers. The ring finger on both hands to be precise. I wondered whether this is normal and I just have to get callouses, like it happens to my fingertips when I play guitar, or whether that means I'm doing something wrong? Because it hurt, more than what I'm used to from accustoming my fingers to guitar again after years of only occasional playing.

The kit is too close to the floor, actually, but I don't have anything high enough to put it on. I'm sitting on an amp that's some 50 centimetres high, and the drums are about at the same height.

Thanks for any advice,

Blisters are normal. Although i tend to get more if the sticks have a "grip" on them. Should fade with time. I'll give you the same advice i gave the other guy on here if you cant splurge for a kit, Go get a used "rock band drum set" from the video game. I saw em cheap at goodwill or im sure you even know someone that isnt using them ha. But you can hack those on your computer to give you drum sounds and you can work on your paradiddles and other rudiments till you can get a real kit. they usually come with a kick and four pads that can be used and you can probably combine it with your current "kit". Also, work on your hand exercises while you dont have the means for other. It really makes the difference.
Yeah blisters are normal.
I get on my middles fingers on the lowest 1/3rd on the fleshy bit. 
Also at the base of my index fingers.
Okay, so I'll just have to build up calluses? That's good to hear. Buying something else… is less good (notoriously broke high schooler here). I think I mentioned it, I'm mainly playing bass and guitar, so investing much in drums isn't my number one priority right now. The drum kit I sometimes borrow atm can be plugged into my PC via MIDI, the thing just, it's old and cheap (something like this), the (cheap cheap cheap) pedals are half-broken, and the snare pad doesn't trigger every 2nd or 3rd hit. Like I said, it gets the job somewhat done, so I'm more concerned about developing the right technique.
Well if it doesnt trigger sounds how are you going to build on anything? You either should develop your stick skills with it turned off  or getting a new kit/machine. I had a drum machine similar to the one you had when i was younger and couldnt play drums all day, you can make beats, learn accuracy and rudiments on them but if sounds arnt coming through i would think you'd have problems bringing what you play on a broken set to a real kit. However, anything is possible.
How tightly are you gripping the sticks and how hard are you hitting? 
I started on a hand me down Simmons electric kit from the 80s. Built a drum rack out of PVC pipe. Worked great.
Harmony: Stratocaster
Alvarez: F-200
Schecter: Omen 6
Fender: BXR-60
Dean: Metalman Z Bass (Betty)
Egnator: Tweaker 15
Pearl: Maximum
Custom: Harley Quinn Bass
Custom: TK-421 Explorer
A steadily growing supply of pedals
I'm hitting pretty hard, I think, and I assume I probably hold the sticks somewhat tight, since I don't really want them flying out of my hand and I don't have any practice holding them.