Germany was planning on resuming unrestricted submarine warfare in the Atlantic - they reasonably assumed that destroying US shipping could bring the US into the war at that point anyway. They ended up staying on plan and resumed unrestricted submarine warfare on Feb 1, 1917. Many US ships stayed in port to avoid the U-Boats, and it did impact shipping. President Wilson attempted to arm merchant ships, but his proposal was rejected by Congress.

If Germany could convince Mexico to tie down the US, it would not only delay US intervention in Europe, but could also reduce the shipment of arms and other supplies since they'd need them at home.

It may have worked out well for Germany had it been possible, but Mexico did not have the means to invade, never mind hold, any US territory. They were already in the midst of a civil war, and the US was also fighting the Mexican revolutionaries led by Poncho Villa.

Even back then the Mexican government also recognized the extraordinary difficulty they'd have holding any of the territory if they did succeed in taking it, not only because the inhabitants spoke English, creating a language barrier, but even back then, the US had the most heavily armed civilian population in the world. There was also plenty of anti-Mexican sentiment in the US at the time that would make controlling the population all the more difficult.

On the other hand, to quote Eduard von Capelle, the German Secretary of State for the Navy: "They [the Americans] will not even come because our submarines will sink them. Thus America from a military point of view means nothing, and again nothing and for a third time nothing."

The American army was at the time the 17th largest in the world, with a bit over 107,000 men. By comparison, the allies on the western front alone had over 4,000,000 soldiers, and on that front Germany had 2,500,000 men. The entire US Army at the time was small, far away, and not at all prepared for major campaigns, so the reality was Germany saw the US as posing little risk, and at that time, the US military really was of little threat to Germany.

The US wouldn't declare war until April, but even then it would not be until the next year, 1918, that US forces would begin arriving in meaningful numbers. By then, Russia was falling apart in a civil war of their own, and the Central powers could refocus on the western front.

As far as the forgery theory, that was an idea at the time when the message was first released. There were a lot of Americans of German descent (as well as those of Irish descent) who did not like the British at all, and who immediately suspected the British of treachery. However, Arthur Zimmerman, the German State Secretary of Foreign Affairs admitted publicly in March of 1917 that he had in fact wrote the telegram, putting an end to all doubt of its authenticity.