No maps, no avatars, no updates, but:
Every time you search breaking news is requested from customer.teliacarrier.com.
Confirmed with Process Monitor, it first makes a request to a subdomain of that URL and then saves a file in
C:\ProgramData\Blizzard Entertainment\Battle.net\Cache\Download
that is called breakingNews.txt which is empty but allows them to interrupt the league queue if needed.
You can confirm this yourself looking at the modified times of the files, the only other things you might encounter that have recently changed (but not when entering the league) are the news images...
I guess the first time after you start the game your league settings are updated to match the league you are allowed to play in; so for example if you play in Practice you will have the slow settings and if you advance to ranked leagues you will have faster settings.
Why: Because I guess this doesn't happen when you load SC2 itself so you can go to single player without waiting for your slow connection. The profile data probably contains less or different data so it isn't updated in that case either. And it should always be made up-to-date when you enter multi-player in case you have played SC2 before on another computer...
Files: Besides creating/accessing some cached .s2m* files,
the only readable file I could find contains this:
<?xml version="1.0" encoding="utf-8"?>
<locale
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="l10n_output.xsd"
locale="enGB">
<e id="1">1v1</e>
<e id="2">2v2</e>
<e id="3">3v3</e>
<e id="4">4v4</e>
<e id="22">2vA.I.</e>
<e id="42">2vA.I.</e>
<e id="52">2vA.I.</e>
<e id="32">2vA.I.</e>
<e id="62">2vA.I.</e>
<e id="23">3vA.I.</e>
<e id="43">3vA.I.</e>
<e id="53">3vA.I.</e>
<e id="33">3vA.I.</e>
<e id="63">3vA.I.</e>
<e id="5">FFA</e>
<e id="11">1v1</e>
<e id="12">2v2</e>
<e id="13">3v3</e>
<e id="14">4v4</e>
<e id="15">FFA</e>
<e id="1001">1v1</e>
<e id="1002">2v2</e>
<e id="1003">3v3</e>
<e id="1004">4v4</e>
<e id="1022">2vA.I.</e>
<e id="1042">2vA.I.</e>
<e id="1052">2vA.I.</e>
<e id="1032">2vA.I.</e>
<e id="1062">2vA.I.</e>
<e id="1023">3vA.I.</e>
<e id="1043">3vA.I.</e>
<e id="1053">3vA.I.</e>
<e id="1033">3vA.I.</e>
<e id="1063">3vA.I.</e>
<e id="1005">FFA</e>
<e id="1011">1v1</e>
<e id="1012">2v2</e>
<e id="1013">3v3</e>
<e id="1014">4v4</e>
<e id="1015">FFA</e>
</locale>