Very nice short answer!! Tru Dat! And remember to raise the eyebrows for those 'high' notes. Plenty of people don't know that and hurt my ears.

I second finding an excellent vocal instructor. You could ask this person to give group lessons too.

If you are singing in a group, decide the exact vowel you are going to sing. It may not be the exact vowel you see, but something that blends better for all, or if you are singing alone, something that is sweeter to your ear. It may be just a little bit of a "tweak" that makes all the difference.

Vocalize on many different vowels.

Sing supported, not on the breath.

It is amazing how much drinking water (at least 250 mls - a cup per hour) has made a difference in my range.

Take instruction from a competent singing teacher.

If they say "raise the eyebrows for those 'high' notes" - please seek out another instructor. To get good notes, the only thing you should raise is your soft palette.

If you're having trouble finding a voice teacher...although these are sooooo helpful to sing with proper technique. Here are a few tips that may be helpful:

Learn to breathe properly. Breathe by allowing your abdominals to expand, then only slowly allow them to return to their original position. A relaxed, engaged breath is the only one that will let you sing to your full potential.

Experiment (slowly) with creating different vowel sounds with your voice by changing your mouth shape, intensity of breath, and thought process (thinking of images/colors/voice types/the sound you're trying to achieve...the list is endless).
As you do this, notice any differences in sound (sometimes recording yourself can help with this...but be gentle on yourself–very few enjoy listening to recordings of themselves), and then try to replicate your thought process to recreate the relaxed sounds you liked most.

As you experiment, allow yourself to sound silly, operatic, and everything in between.

Notice areas that you're keeping tension when you sing (this is often in the jaw or neck area, which strains the muscles of your vocal mechanism). Sometimes it helps to massage your neck/face or lightly stretch as you sing so that you can properly relax these areas.

Find your breaks (those notes where your voice wants to "crack," has a "fuzzy" consistency, or is just generally less comfortable). Sing with lighter over them, with more space at the back of your mouth (your soft palate), and allow your breath to move freely as you sing them (think to yourself, "I am a wind tunnel").

A lot can be done from good (emphasis on good) videos. A group of men from Zambia singing in their (Anglican?) churches learned entirely from copying what they found online; Anna Lapwood of Pembroke College, Cambridge goes to visit them and works with them in person, and she’s hosting one soon for his own recital.

Raising the soft pallate is definetly the way, but I'm struggling with how to push the air. If I sing "whispering" (just sing very quietly), everything goes well, but there is no much power and, of course, no one can hear. When I go louder, I need to scream on the higher notes and when I fix this, I start to blow the notes and the tone and texture are getting bad. I don't even need to go extreme with this Alleluia, just with the Da pacem, Domine, in the key from this Giovanni Vianini's execution, I'm having this problem in 'Do-mi-ne' part.

About the vocal teacher, could it be any classical-repertoire good one? I felt that the technique for gregorian chant is somewhat different from classical music. But, anyway, I would like to have in-person classes in the future, because now it will be very difficult and there is no much good teachers avaliable.
Is there any online course, or even any method book that I could follow?

Soneto -- Your entire body is a stereo speaker. So is your head. When you were a baby, you could fill an entire concert hall with sound, and it probably took a really long time to run out of breath. So you don't need to push anything.

So basically, you get louder by relaxing, by letting the sound sort of flow out more, and various unconscious things will happen to make things louder. When you get a good result, remember how it felt, and see if you can do it again. Don't be afraid to make silly sounds.

You can also do things like visualizing the sound, coming out of your mouth, going back into your head at various points right above your nose/eyebrows. Play with this, and see if it helps. If it doesn't do anything for you, just forget it.

Have a look at diaphragmatic support, proper posture and lip trills. Can't really go wrong here, this is universal for all kinds of music. Here's a couple videos.

https://youtu.be/wIhnMq-rzYc?si=x8q1JJwt2f-D48t6&t=60 - Proper posture, breathing exercises, and a short tutorial on lip trills.

https://www.youtube.com/watch?v=VyMb-hNYxh4&t=149s - Bit out of context from the video, but pretty good examples of common pitfalls when doing lip trills.

"Perhaps you are trying to sing above your natural range."

That certainly can be true in SATB part-singing: the voices of majority of women are mezzos, and the voices of a majority of men are baritones, and for untrained singers SATB parts can be a burden on those voices unless the parts are divided further, et cet. True soprano, alto, tenor, and bass voices are rarer than mezzo and baritone voices on the ground. Directors who are unaware of that reality may ignore the signs of it.