Did American imperialism in the 1890's combined with the American expedition into Mexico after Pancho Villa make the Mexican government legitimately afraid of America annexing Mexican territory?

The United States was unlikely to actually annex any additional Mexican territory. Since the presidency of Roosevelt the United States had made a concerted effort to not appear like European Imperialist colonizers and conquerors. Cuba, conquered from Spain in 1898, was given full freedom in 1902. Guam, the Philippines, and Puerto Rico were held as territories, not annexed as states. The three year war with the Philippines from 1899 to 1902 showed how unpopular and difficult wars of conquest were in the 20th century United States.

What the United States was likely to do was destabilize Mexico and turn it into a United States Protectorate like Cuba, then permanently occupy Veracruz so as to control the oil fields there. Had they done this it would have been playing into the Germans hands as it would have distracted the US from Europe. I guess the answer to your question is that the Mexican government legitimately feared the US would do something which would deprive them of money, power, and resources.

Why didn't America recognize the Mexican government diplomatically before the Zimmerman telegram?

They did. The United States always recognized a faction throughout the Mexican Revolution.

Was the Mexican government even capable of any sort of large scale offensive?

No, and Mexico knew it. The best thing Mexico would be able to do is distract the United States for a few months. That is all Germany thought they needed to knock out France and cripple the UK so as to force a negotiation and end the war with favorable terms.

Were the Germans inspired by the success of the British inciting rebellion in the Middle East?

Not that I am aware of. There weren't many Mexican nationals in the US western states that were simmering with rebellious rage or planning insurrection. The promises of returning western states to Mexico were just an enticement.

Did the German diplomats think that the Mexican government would actually agree to the proposal?

Yes. German diplomats and advisers had been actively promoting war between the United States and Mexico since 1914. They continuously funded efforts to draw the attention of the United States and distract them from the war in Europe. I don't think Germany really thought Mexico could win, they just wanted Mexico to occupy and distract the US. The Germans were successful in a way, as the ongoing tension between the United States and Mexico resulted in Mexican neutrality and a guaranteed trading partner.

Any additional reading you might have would be greatly appreciated. Especially books to get a decent overview of the diplomatic situation between Mexico and America during the time period.

For general reading I'm a huge fan of the Oxford histories. The Oxford History of Mexico is good, and I think tackling the situation from Mexico's point of view will give you a better insight. Other than that you might check out From Colony to Superpower to look at US Foreign relations in the time, but if I understand correctly it is about to get a new edition titled Years of Peril and Ambition that just focuses on the time from 1776 to 1921 instead of to the current.

While I can't fully comment of the German perspective, I can give you some context from the Mexican side.

Diplomatic relationships between Mexico and the United States had been in a state of permanent tension since the events of the Decena Tragica (The Ten Tragic Days). In February 9, 1913, a group of Mexican military leaders aligned with the democratically overthrown government of Porfirio Diaz decided to state a coup against then President Francisco I. Madero. The coup had important support from the military sectors of Northern Mexico, who identified Gen. Bernardo Reyes as the legitimate successor to Diaz. Reyes died on the first days of the coup, leaving the movement without clear leader. The US Ambassador to Mexico at the time, Henry Lane Wilson, saw the opportunity to help a dissident military faction inside Madero´s administration, leaded by Gen. Victoriano Huerta. Lane Wilson saw the increasing talk among government officials regarding new regulations and possible nationalization of the Mexican oil industry as a serious attack to American national and private interests, and so a government favorable towards the US was deemed necessary. Huerta, with the help of Lane Wilson, played both sides for a few days, until the opportunity presented itself for him to denounce Madero on the grounds of incompetence to put an end to the conflict. Madero resigned on February 19 and was killed two days by Huerta's officials, even after being promised immunity on exile.

These events helped to fuel anti-imperialist and nationalist sentiments among the country's middle-classes and anti-Diaz military, who valued the limited but growing political liberty that came to be with Diaz' regime change. The participation of the US Ambassador greatly strained diplomatic relationships between both countries, as not only the government but public opinion as well started seeing the United States as a real threat to national sovereignty.

The fear wasn't unfounded. The Mexican-American War (1846-1848) had been part of the nationalist discourse for decades, even on Diaz' administration, where diplomatic ties improved greatly thanks to American investment on diverse economic industries considered priority by the regime. The war not only meant the loss of half of Mexico's territory, it also represented the practical consequences of American imperialism and the origin of Mexican distrust for its northern neighbor. The participation of Henry Lane Wilson on the coup against Madero revived a lot of sentiments among the general populace and shaped the discourse of many political actors who would become important players in the Mexican Revolution.

Relationships heated once again on April 1914, when a group of American sailors were arrested by the Mexican government for entering the Port of Tampico without authorization. The arrest itself quickly resolved, with the prompt release of the sailors and an issued apology, but Woodrow Wilson's administration, having previously denounced Huerta's government as illegitimate (as well as dismissing Ambassador Henry Lane Wilson shortly after taking office in regard to his actions on the Madero's coup), decided to seize the opportunity as justification for American intervention on the Port of Veracruz. Wilson had previously received intelligence on a weapons shipment, coming from Germany, in support of Huerta's government heading to Veracruz. Having previously denounced him as illegitimate, Wilson openly supported the cause of the Constitutionalist Army, headed by then Governor of Coahuila Venustiano Carranza, to overthrow Huerta. As to help the Constitutionalist cause and protect American citizens residing in Mexico, Wilson ordered the occupation of the Port of Veracruz on April 21, 1914.

The occupation, even though supposedly meant to help the opposition cause, was seen in the country as another episode of American interventionism. Important protests against the occupation were held in diverse urban areas, including Mexico City. The event was heavily criticized on the press, while both Huerta and Carranza publicly denounced it. The conflict resolved on November 1914 with the signing of peace in the Niagara Falls Peace Conference between President Wilson and the just victorious Venustiano Carranza. Among mixed intentions, the conflict brought US-Mexican relations to an all-time low, creating mistrust and discontent on both countries with few hopes for the near future.

Experiencing such a context, it is clear why the Germans saw conditions for such a deal. By the time the Zimmerman Telegram was sent, Venustiano Carranza hadn't officially taken office as President of Mexico (Having ruled de-facto as Head of the Executive Power-Chief Leader of the Constitutionalist Army since 1914), having just finished a three-year violent power struggle among opposite revolutionary forces in the Revolution. The state of the Mexican Army was dire, as resources had been depleted in the armed conflict and the first proper government in years was months away from properly taking office (On May 1, 1917). The telegram was intercepted and published in the United States on March 1, 1917, two whole months before Carranza was properly recognized as President. While mere speculation, had the telegram not been intercepted by the US before reaching Carranza it is extremely doubtful the government would have acted on it. The conditions to take a proper offensive just didn't exist at the time, and the government wouldn't have risked the fragile stability it had to seek a mostly wishful proposal.

In short, the German government was trying to appeal to Mexico's nationalist sentiment without regarding the simple unfeasability of the proposal. A Mexican attack on the United States amounted to a shallow distraction, as the capabilities of the Mexican government at the time were almost non-existent. In any case, the interception of the document created generalized outrage in the United States, prompting Wilson to declare war on Germany. The telegram not only was impossible to act upon, it ultimately backfired.

Sources and further reading:

• The Mexican Revolution (1986) by Alan Knight (Hands down the best account of the Mexican Revolution in English, massive study but well worth the time)

• El Ejercito Mexicano: 100 Años de Historia (2014) by Javier Garciadiego (Spanish)

• Carranza, El Último Reformista Porfiriano (2009) by Luis Barrón (Spanish)

• Febrero de Caín y de Metralla, La Decena Tragica, Una Antologia (2013) by Antonio Saborit (Spanish)

• Historia Diplomática de la Revolución Mexicana 1912-1917 (1958) by Isidro Fabela (Spanish)

For a really fantastic account of the Mexican-American War (1846-1848) from the US perspective:

• A Wicked War: Polk, Clay, Lincoln and the 1846 US Invasion of Mexico (2012) by Amy S. Greenberg

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Is there any case to be made for the Zimmerman telegram as a deliberate attempt to sour Germany's relationship with the US?

It seems like this happened at about the point where Germany's choices were either a negotiated peace or to roll the dice on unrestricted submarine warfare. Doing something really outrageous to the most logical mediator of a peace treaty seems like a pretty good way for people in the submarine camp to stack the deck in their favor.

Obviously without any kind of evidence this is basically a conspiracy theory, so is there any evidence to back up this view?

You may want to check out Frederich E Schuler's Mexico Between Hitler and Roosevelt. Schuler teaches at PSU (where I am working on my undergrad) and the course I took with him gave me the impression that he has a good grasp of Mexico during WWI.