1) Assuming you connect from home to foo, you need a reverse tunnel (-R)
ssh -R 8080:localhost:3000 foo.mycompany.com
This will enable processes running at foo to connect to localhost:8080 and actually speak to your home computer at port 3000. If you want other computers at your work to be able to connect to foo:8080 and access your home computer at port 3000, then you need
ssh -R 0.0.0.0:8080:localhost:3000 foo.mycompany.com
but for this to work you also need this option to foo’s sshd_config
GatewayPorts yes
2) The best way to create an http proxy with ssh is with socks. First connect with
ssh -D 8888 foo.company.com
then go to your browser connection settings and enable proxy connection, choose socks4/5 and host: localhost, port 8888. Then just type http://bar.mycompany.com in your browser’s address bar.
3) Now you need a local port forward (-L).
ssh -L 3333:db.mycompany.com:3306 foo.mycompany.com
This means that you will be able to connect at localhost:3333 from your home computer and everything will be forwarded to db.mycompany.com:3306 as if the connection was made by foo.mycompany.com. Host db will see foo as the client connecting, so you need to login with the same username and password you use when working from foo.
Adding -g flag will enable other computers from your home network to connect to your computer port 3333 and actually access db:3306.