Semps are winter-hardy, best grown outdoors year-round. Those are not buds but "chicks" - new plans. There is not need to separate them unless you want plants growing in different spot...They look great - at least to me - if all bunched up :)

2 sprouts are too small to ID - post when bigger. I see another hardy succulent in your last photo. They need lots of light - same as most succulents.

Whoaa.... :O Those succulents look amazing, what is on the second picture on top of them, it looks like they were sealed with spider silk. :)) Thank you Rina! Are you a botanist? You know everything about any plant :))
I've let the sempervivum in the same soil it came with and completed the rest with universal soil when i moved the plant in another vase ( it has drainage holes ) and at the base of it i've put a layer of sand... i know that they prefer more soil mixed with pumice, but i can't find that in my town.... Here it's not such a great weather in winter, now it's more like autumn - with more humidity in the air so i prefer to keep the plant inside.
The sprouts as you see them, the longer one come with the pot and that made me curious as what it is, the little one formed by 2 leaves comes from one stem ( or i don't know how to call it ) from the sempervivum ; and the other succulents in my last photo are just some tips who were cut during transportation of the package of this one

2nd pic are Sempervivum arachnoideum. Yes, they look like they have spider webs all over :)

Don't worry about pumice - I can't find it easily either. Can you get perlite and/or chicken grit? I would suggest not to use sand, and especially not in layers. Unless it is very grainy - particles at least as big as a dressmakers pin head :); well, they could be bit smaller but not much. I wouldn't use anything that falls thru kitchen sieve.

Your last photo is another hardy succulent - looks like Sedum rupestre Angelina.

The little sprout you said is growing from the stolon (that is the 'stem' baby plant is attached to mother plant) is probably another offset. But it is extremely etiolated - that means not enough light. I can't tell for sure what other plant is - again, it is etiolated and should get much more light. Post when it gets bigger to ID.

Could you tell us what is your general location? Can't figure out where ..."Here (it's not such a great weather)"... is, haha...Best if you put that info next to your name - see how others do - you can edit your profile to do that.

Thank you for the compliment! I am not a botanist...just growing plants as a hobby. I have learned a lot since I came here - many ppl gave me great advice.

Ah! I tought you were :)) but actually i think you know as much as one :))) you give back the knowledge you've received and you care for other plants too, it's very althruistic :D The arachnoideums looks so interesting, aren't they soft at touch? :))
You were right with the other succulent that i have, that was the name, you have such a sharp eye :)))
I've already used fine sand when i've reppoted the succulent, but i used only one layer, at the base of the vases...when the plants would get a little bit bigger i'll move them with a proper soil... In the meantime i'll try to find perlite or chicken grit, but i don't have much hopes because i live in a small town and growing succulent here ( in the Southern Part of Roumania, Europe ) it's not so popular, so you can't find everything what they need, but maybe i'll have a shot with chicken grit... i hope on this one.
stolon - i've got it now, thank you :)) ( and for the other advices too, i've learned a lot from you )
I would post another picture when that little sprout gets bigger... but in the case of the light, i've put all the succulents on the most light up place in the house, but i guess they need more hours of light since it's winter and days are shorter...should i put a lamp for them?

Also...i've wanted to ask about this one too ( i've bought more succulents, i like them, i hope i can take care of them too :)) )
I've noticed on this one that it gets roots from near the leaves ( which it seemed odd for me because i've never seen that happen at another plants that my mom have :)) ) Should i move Sedum Album Coral Carpet on another pot... with wider space?

Last one is Sedum rubrotinctum - it also needs more light. Here is recent thread about same plant; click on it see many good photos. It grows lots of aerial roots - hair-like roots from anywhere on trunk, do not worry about them.

Many of us have same problem with not enough light during winter, and use supplemental lights. If you could get some, I am sure you'll see difference in way your plants grow and also in their color. Can you get CFL or LED lightbulbs? Those are very efficient = use much less electricity than regular (incandescent) lightbulbs. Fluorescent lights are also very good, but if you get some make sure they are either T8 or T5. Older types (T12) are not very efficient, so don't waste money on them. There are also many LED grow lights available, that supply lots of red and blue for plants, but I don't have any of those and can't advise you. And they are more expensive to buy.

Potting mix is quite important for growing succulents successfully. Where I live, chicken grit is available in farm supply stores and quite inexpensive. Maybe you can find some - it is sold to be given to chickens :) Perlite is also easy to find here, hope you can find some. Getting just perlite would help a lot; mixing it 1:1 with soil. In the meantime, make sure you water only when soil is almost completely dry deep in the pot.