Replacing needle bearing threaded headset with an ordinary one I got a bicycle like this one (90's trekking/city bike). When I got it, the headset seemed overtightened, when I took it apart I saw that it was somewhat damaged (looks like ridden for some time while being too loose). I noticed that the bottom bearing is needle, upper was regular ball bearing. I managed to get it together and tighten it just right, but I think it should replaced anyways.

Will I be able to replace it with ordinary ball bearing based headset (I guess I won't be able to find the same ball/needle set easily)? Are head tubes for threaded headsets all the same?
In replacing a threaded headset on an older bike, there are two other considerations we need to make beyond selecting between ball bearings or needle bearings (though, IMO needle bearings are ideal for a number of reasons that I won't get into.)

First, you need to measure the stack height. Stack height is the amount of space the headset takes up, not including the parts that fit inside of the head tube. If you measure the entire length of the steerer tube and subtract the height of the head tube you wind up with a difference that is taken up by headset cups, bearings, the keyed washer, and the lock nut. You can add or subtract spacers to make fine adjustments.

To measure the stack height, simply measure:



You'll wind up with a measurement in millimeters. Try to find a new headset with a stack height equal to or less than what you have. If there are spacers installed (smooth or knurled rings between the top lock nut and the upper race) ignore them, but be sure to include the height of the lock nut and keyed washer (just under the lock nut.) If you wind up with a new headset that is just slightly too tall, you can omit the keyed washer if you use some thread locking compound on the lock nut, but it would be better to just get the right dimension in the first place.

The second thing you need to do (and this will be more difficult) is to verify that you have an ISO headset (as opposed to a JIS headset... more extensive information about different headset measurements and how to adjust them Sheldon Brown and Headsets here.) ISO is the standard size, but JIS headsets were used on lots of Japanese and Taiwanese bikes from the 70s until the early 90s. There are many other sizes, but for most 90s bikes this will suffice. 

To finally get to your question...no not all bicycle headset frame cups are identical. Another measurement, the diameter of the fork crown seat, also varies between standards. Two common threaded headsets you'll run into are 1" ISO and 1" JIS and they are ever so slightly different. If you have access to a lock nut wrench and metric calipers discerning between the two will be pretty straightforward. 

Remove or loosen the fork in the frame enough to see the fork crown race when you push the fork down and lift up the bearings and retainers. You need to measure the inner diameter of that fork crown race.



The fork crown race in the photo is bronze colored, while the seat at the bottom of the steerer is silvery. The seat is slightly larger than the steering tube, so measure right where the seat and the race meet. In an ISO headset that measurement will be 26.4 mm. In a JIS headset that measurement will be 27.0 mm. I will tell you from my own experience that the two are not interchangeable. Additionally, the frame cups for an ISO headset are 30.2 mm in diameter while JIS cups are 30.0 mm. It is much easier to fudge installing 30.2 mm cups in a frame designed for 30.0 mm cups, but have a shop do it (and only in a steel frame.) Finding cheap JIS headsets is easy enough and, like I said, ISO is pretty standard.

One final note: replacing a headset is actually pretty doable for most people so long as they're careful and reasonably clever. If you're unsure about any of the steps you should take it to a shop. It is a quick installation and relatively inexpensive. They have all of the right tools which will make the process far easier than using homemade tools.