Remake and get aesthetic pleasure or be guided by the principle of "working - do not touch"?

Question

Network administration

Good day, dear experts and involved. Help to understand the situation.
Probably the situation is typical and banal, but still.
I got a part-time job in an educational office. The staff of cars is about 100 + comp. class.
The level of IT skills of the staff is not high (at all), but the needs are small. Physically the network is laid and works normally.
There is a server that filters traffic and acts as a gateway to the Internet and it contains a shared folder for network users. On one more machine on the virtual machine, web filters are installed to filter requests from the computer class (all sorts of bad words) Also, some powerful server will be ordered now for 200 thousand.
All this seems to be functioning normally, BUT I have some feeling that there is no system, some kind of discord and vacillation that suits everyone, out of ignorance.
Namely, the list of what does not suit me:
1. User behavior on computers (one local disk, a bunch of left software, everything is stored on the desktop. When I see such PCs, I remember the feat of Hercules with the stables.)
2. Why do we need a powerful server, if it is used stupidly as a gateway and usergate is on it?
3. No network communication programs between users (all calls on a mobile phone)
4. Long mail work, everyone is patient and waiting.
5. The site is on some free hosting, not on a server.
What I would like on an intuitive level.
- centralized update of anti-virus databases
- centralized storage of mail (so that users through outlook do not get to mail.ru, but to the server, and the server itself picks up mail from soap every few minutes)
- a ball on a separate PC, not a server - a
centralized web filter
- transfer the site to our server.
-Do you need a domain?
And actually I ask for advice from experienced people: I'm getting to the bottom of the situation. newbie or does it make sense to try to reorganize the network?
I myself am a programmer, I decided to consider networks for gaining experience. (This information is in order to warn against exclamations "you don't know how, don't take it, not everyone can be an administrator" and so on)
Thank you very much for reading and for the advice.

Answer 1

Measure seven times, think, make sure that this is the amount of work for many days that no one will pay you.
And leave this idea.