There are a couple of ways to solve issue 1.
Figure out the name of the image that's in there (sorry, don't know it off-hand) and go into the master page file and change the src URL to the location you're storing the picture you want to put in there. You may need to upload it to your SharePoint farm first before you set that up, depending on whether or not you've already placed the image somewhere people can reach.
Do the same as above, only instead of changing the information in the masterpage, swap out the actual file. I don't recommend doing this because there's a good, solid chance MS will copy the original image back in a later CU or service pack, but sometimes the masterpages can be tricky themselves.
Your second issue sounds to me like a problem with the page layout itself, which can be a bit of a pain to edit. If you can open the page in question in Designer and resize the Web Part Zones themselves through there, that might work, although I don't think you'll be able to (sorry, can't check from where I am right now). Your other option would be:
- Go into the document library that page is residing in and see which layout it's using.
- See if you can then go into your Page Layouts directory in SP Designer (for your root site of the site collection in question) and make changes there. Make SURE you copy and save it before you make any changes, because depending on the size of your farm there might be a lot of pages affected if you happen to break that layout, and, well, Microsoft assumes you're not going to be touching those.
As with #1, this is not a perfect solution; since you're tampering with parts of SharePoint that MS considers to be "core", it's entirely possible that they'll undo all of your changes in a future CU/SP.
The Useful Links web part... may or may not have its own separate div id that you can take advantage of. If it does, adding style ought to be as easy as inserting a Content Editor web part somewhere on the page and adding some CSS into that. You would want to make that part hidden from view, of course...
As an alternative to changing existing stuff for 2 and 3, you might want to think about just creating an entirely new page layout and using that for your home.aspx or default.aspx pages (you may need to delete the original). That would allow you to retain the original page layout but have your own with all of its own stuff in it that Microsoft will never, ever go in and edit.