Greetings,
Welcome to the USCF.
I don’t think there is a single FREE product that will do everything you mentioned, but if your willing to try several products, and see what fits with you, you can do all that with free software.
PGN files are plain-text ASCII files in a particular format, they can be created in any text writer, but you have to follow the rules of the format or the files won’t work in anything (databases and chess engines)
Creating a PGN file is fairly easy with a PGN reader/writer. There are many around, I think that easiest and most comprehensive is “Winboard”.
chess.com/download/view/winboard-427
You install the program, one of the shorcuts it creates is called “game viewer”. Click on this and you see a chess board at the starting position. Take your game score sheet and play in the moves by moving the pieces on the board. When your done, click “file” and then “save game”, pick a folder and a file name, do yoursef a favor and pick a meaningful file name. You have your PGN file. Whatever datebase or chess engine you use will import this file. As you are playing though the game (or later as you are going back over it), you can insert notation comments by clicking “mode” and then “edit comment”. What you type in will be saved in the PGN file. Cicking on “mode” and then “edit tags” will allow you to insert header information. Winboard can also be use to connect to chess servers and play online, though over the years most chess servers use dedicated (must use our’s) software. Winboard has a chess engine built in, but you can only play against the engine, not use it fo evaluation of games.
One of the problems I have with Chessbase is that it has it’s own file format. Yes, it will convert back and forth to pgn, but I personally found it querky and non-intuitive. But, it will only cost you your itme and effort to find if it works for you. The pay version is more comprehensive.
Another free chess database is Chess DB.
chessdb.sourceforge.net/
Another free porduct to check out is BabasChess.
babaschess.net/
Like Winboard, you can use it to connect to some chess servers. It will also read and create PGN files. It has a chess engine built into it that can be used to evaluate your games, but not to play against it (the opposite of Winboard).
A lot of this depends on how comfortable you are with computer software.
Good Luck.