Doug,
Here's my opinion about USD.
USD is a dominant currency within the global market. Most transactions (both between countries and global corporations) in the global market are conducted in terms of USD.
Most, if not all, countries have most of their reserves allocated in USD, because exports and imports between countries usually take form in USD.
So, with such significant role in the global market, USD surely effects the market.
I am involved in FX market, so I explain it from FX market perspective.
Major currency pairs traded are of USD-related. Example: USD/CHF, USD/JPY, EUR/USD, GBP/USD, AUD/USD.
If USD strengthens, USD/CHF and USD/JPY move up, and EUR/USD, GBP/USD, and AUD/USD fall.
Thus, traders may position themselves in long positions in USD/CHF and USD/JPY while position themselves in short positions in EUR/USD, GBP/USD, and AUD/USD.
I don't understand what you mean with "when the markets go up does the dollar go down?"
Which market did you refer to? Stocks? FX? Treasuries? or commodities?
As for why some companies prefer weak dollar, well, this usually applies to US exporters. Weaker USD means cheaper price of exported US goods. If US goods priced too high, there could be limited market due to limited purchasing power of the world.
On the other hand, if the USD is weak, US can import less amount of goods from abroad given the same amount of money compared with the times when USD is stronger.
US Treasury has stated its 'strong dollar' mantra repeatedly. This, however, I don't really certain about.
Suppose they DO want a strong dollar, why would Bush's Administration lobbied China and Japan to ease their stances regarding their FX policy?
If China let loose their FX pegging of CNY, and Bank of Japan stop intervening, surely the USD will fall afterwards.
It appears to me, US Treasury still holds its strong dollar policy-in GENERAL, but not against CNY and JPY.
Right after some futile efforts to persuade the Chinese government to let loose of CNY peg, the US proposed quotas on Chinese textiles (clothings) and TV sets. Perhaps, I think (just my opinion), as an effort to 'punish' China.
Add the EU-US row over steel, all in all, these have created more woes to the USD, since these rows could ignite a global trade war, which in turn, will disrupt the balance of global economy.
Hope this helps, Doug.
Cheers!