I want to replace my bicycle's rear wheel. The existing wheel has a 3-speed threaded freewheel installed. The new one will have more cogs. I'd like to know if there could be any problem fitting a larger freewheel on the frame. I know that I have to check the hub's length. This has to be equal to the distance between the frame dropouts? Or it is better to be a little smaller and if yes how much? The dimensions of the hubs are standard or each brand has different lengths?
Thanks in advance!

If you have a bike with only 3 speeds on the freewheel it is rather old.
true, better to be on the dot, or a few MM short.

if you cannot supply pictures to accompany the post,
you must visit a Bike shop that can see what you are speaking about.

More gears..may be simpler to use an internal gear hub, built into a similar wheel,
than do the extensive spreading the frame may require.

Hubs are standard width, which varies on intended use. For instance a road bike rear hub is a few mm narrower than a mountain bike hub.

If you are able to replace the freewheel to a 5/6/7 speed you will also have to change out your shifter and derailleur. There is a slight chance you can use your existing hub with a newer freewheel with 5 to 7 cogs.

By chance is your bike a Russian dating from the mid 1970's? I remember putting several of those together for the bike shop I worked at then.

Im actually doing this now... you just have to be careful to make sure you have enough space on your thread so that the end of the freewheel doesnt hit the dropout part of the frame if your adding more cogs it might be better to try a couple on at the lbs to get a better idea as to what will work or not...

It could be made in Russia. It is a Peugeot and I am not sure about the model. I have started a thread in order to find some help to identify the model. It has to be a mid 1970's PNC40 or PE40, not sure yet.

I don't think it could be such a problem, cause the cogs are getting smaller as they reach the dropout.

The hub (with axle and freewheel installed) should fit exactly between the dropouts or it should be better to leave a small gap on the left and right?

I think what he meant was, that putting on a freewheel with more cogs, may cause the freewheel to extend past the end of the axle, or so close to the end that the smallest cog/or chain will rub the inside of the dropout. Perhaps an extra washer or 2 can be added to the axle then redish the wheel a bit, to make it work. All depends on the actual specs, so it's just wild guessing so far.

as long as you can get the drive-side locknut to clear the end of the cluster it will not bind in the dropouts. the chain may require a little more clearance. use spacers as necessary, then dish the wheel. if you have minimum clearance and dishing proves to be unsafe in your opinion, you will need a longer axle and a larger dropout width. which may or may not require the use of a different frame.