Ok, so 900K rows isn't a massive table, it's reasonably big but and your queries really shouldn't be taking that long.
First things first, which of the 3 statements above is taking the most time?
The first problem I see is with your first query. Your WHERE clause doesn't include an indexed column. So this means that it has to do a full table scan on the entire table.
Create an index on the "data_updated" column, then run the query again and see what that does for you.
If you don't need the hash's and are only using them to avail of the dark magic then remove them completely.
Edit: Someone with more SQL-fu than me will probably reduce your whole set of logic into one SQL statement without the use of the temporary tables.
Edit: My SQL is a little rusty, but are you joining twice in the third SQL staement? Maybe it won't make a difference but shouldn't it be :
SELECT temp1.element_id,
temp1.category,
temp1.source_prefix,
temp1.source_name,
temp1.date_updated,
AVG(temp1.value) AS avg_value,
SUM(temp1.value * temp1.weight) / SUM(weight) AS rating
FROM temp1 LEFT JOIN temp2 ON temp1.subcat_hash = temp2.subcat_hash
WHERE temp1.date_updated = temp2.maxdate
GROUP BY temp1.cat_hash;
or
SELECT temp1.element_id,
temp1.category,
temp1.source_prefix,
temp1.source_name,
temp1.date_updated,
AVG(temp1.value) AS avg_value,
SUM(temp1.value * temp1.weight) / SUM(weight) AS rating
FROM temp1 temp2
WHERE temp2.subcat_hash = temp1.subcat_hash
AND temp1.date_updated = temp2.maxdate
GROUP BY temp1.cat_hash;