As of right now, phpBB doesn't support either Tables or HTML (for security reasons). I will look for some solutions today and report back later.

HTML can't be enabled because of security reasons but I have implemented some new BBCodes that should allow you to do everything that you need to do. There are buttons on the toolbar above the posting box or you can type them manually. You should be able to create your HTML table as normal but replace <> with []. Cellpadding is automatically set to 2 with a border of 2 and 0 cellspacing. If you want to add css styles to a table element, you can just add them in like normal without the style tag and the tag will be created.
Sample:
[table][trbackground:#002356;color:#FFF;][td]Test Column[/td][/tr]
[tr][td]Test Data[/td][/tr][/table]
BBCode (Remove * to make it work):
[*table][*tr background:#002356;color:#FFF;][*td]Test Column[/td][/tr][*tr][*td]Test Data[/td][/tr][/table]

Just a note with this. If you create a table, put all of the BB Codes on 1 line. Otherwise it will push the table a ways down the page. We are looking for a solution to this problem but until one is found, this is what we'll have to do.

I found a way to enable html in posts that should be secure. You should be able to use it now. If we run into any problems then I will have to disable it. You don't have to do anything special to use this functionality...just add it to your post. The only tags that are allowed are table, tr, td, th
<table border="2" cellpadding="4" cellspacing="0">
<tr style="background:#002356;color:#FFF;">
<td>Hello</td>
</tr>
<tr>
<td>World</td>
</tr>
</table>

You will still want to keep your html on 1 line unfortunately.

I'm not aware of any reason that it wouldn't work. I will have to dissect the CSS and see if there is something in there that could be messing it up.

I believe we have a solution here. Cellpadding, Cellspacing, Border, and Style should all work with HTML tables. Also, they don't have to be condensed to 1 row anymore.