What parts should I buy?

I would love some help on deciding which parts to buy.

Background: I have always loved to commute by bike. Unfortunately I neglected my poor bicycle aside from some basic maintenance. This has resulted in my chain starting to skip and to rust. However I have decided to better myself and give my poor bicycle the loving care it deserves. Unfortunately I am having trouble deciding on what needs to be done and what parts to get.

My uneducated problem analysis:
– The chain is skipping teeth and has started to rust
– The sprockets are worn down
– The large and the medium sized chainrings are worn down (unevenly)
– My bike is missing a chain guard and I am unsure, whether or not I can install one

The previously installed parts (from what I can gather, I bought the bike used):
– Shimano Deore Mega 9 drive train
– 3 Chainrings (44 – 32 – 22 teeth respectively) with the print "Hollowtech" with four bolts. The outer chainring has a spacing of ~74mm between adjacent bolts, which would suggest a BCD of 104mm (which is supported by this website).
– 9 sprockets cassette (32 – 11 teeth)

The replacement parts I would like to buy:
– Chain: Shimano Deore CN-HG53 bike24
– Cassette: Shimano CS-HG400-9 (11 – 32 teeth) bike24
– Chainrings: Shimano Chainring – 3×9-speed – both 44 and 32 teeth bike24
– BLUECHAIN Freewheel Remover for Shimano bike24
– Topeak Universal Crank Puller bike24

I already own a chain breaker tool.

My questions for you:
1. Do I need to gather more information on the installed parts?
2. Do I really need to replace all the parts listed, since this is a rather expensive grocery list?
3. Expanding on that: Is a crank puller really necessary for replacing the chain wheels?
4. Are all the parts listed compatible with my bicycle?
5. I am especially worried that the chain rings and crank puller listed are incompatible with my bicycle
6. Do I need additional parts for the repairs?
7. Are all the parts of appropriate value and quality for the money (like I said I mostly use my bike for commuting)?
8. Can I just use the old chain instead of a chain whip?
9. Should I additionally buy grease for the bearings?
10. How do I know, whether or not a chain guard is compatible with my bike? I couldn't find any good resources on that, but looking at my bike, it doesn't seem that way
11. If not can I install a chain ring guard instead?
12. Can recommend any good, reputable sellers here in Germany?