connect to ssh server thru 80 via HTTP proxy?

PuTTY supports tunneling SSH through an HTTP proxy. See this guide

Yes, corkscrew is useful. See this tutorial.

You are unlikely to find any publicly accessible solution, though you could setup one for yourself fairly easily if you don't mind paying a few $ per month. Get the cheapest VPS you can find ^{_{(this is the cheapest offer in the first page of WHT's VPS advertising area, you may find cheaper if you search deeper and are willing to pay quarterly or annually)}}, make sure no web servers are installed, installed rinetd, and have it it listed on port 80 forwarding connections to your home server's SSH port. This requires no new software at either the client or server end but does require a new external machine with a little cost.

Your corporate firewall may do more than just block ports though - if they are using packet inspection it may block your SSH attempts as it could identify the connections as not HTTP.

Depending on what you wish to do with your SSH connection you could install something like this script on your web server. Obviously this would not allow port forwarding and such, but it would be fine for many uses. The traffic traversing your corporate firewall will be plain HTTP(S) connections so should pass even if packet inspection or other protocol limitation techniques are used, though if you use HTTP rather than HTTPS your then your traffic is not protected by encryption so your corporate overlords will be able to record the SSH sessions if they wish to do so. Also, make sure you are aware of relevant security matters configure any such script correctly - you don't want to open a route for anyone to get into your server unauthenticated...

Before considering any of the above give careful consideration to the fact that, as other have pointed out, circumventing security measures is likely to be a serious enough matter for your employer that your job is at risk if your actions are noticed.