I just want to learn a little more about an area of US History I am unfamiliar with.
Reagan's status as an icon of the modern Republican Party is due to a number of things. The party views him as someone who held true to conservative values while still being able to grow the party and oversee a resurgence of American economic and military power. There is merit to each of these, although modern politicians tend to overlook the pragmatism of Reagan in office and opt to conflate the rhetoric with results while ignoring the way he actually did business.
Reagan was an exceptionally popular president with both Democrats and Republicans. He left office with an approval rating of over 60%. During the elections in 1980 and 1984 he received 26% and 27% of the vote from registered Democrats and throughout his term of office relied on Democrats to break with their party and vote for his agenda. The so-called "Reagan Democrats" were largely white, blue collar workers from the Rust Belt who felt economically and culturally left behind and bought into the optimism of Reagan's message and rhetoric which was unabashedly conservative. Modern Republicans point to this and identify Reagan as a coalition builder to be emulated and argue that it means the conservative message has a broad appeal and there is little reason to change their tone. This tends to ignore the fact that the Reagan Democrats largely did not remain in the party and many felt bitter that they did not enjoy the same economic growth that the rest of the country did in the 1980s.
The economic growth of the 80s is the primary reason the wing of the Republican Party comprised of businessmen still revere Reagan. Reagan entered office amid an economic downturn and with the stigma of the 1970s stagflation still looming. While the first two years of his presidency saw continued depression (and his approval bottomed in 1983 at 38%) the rest of his time in office was one of remarkable growth. Reaganites credit the implementation of supply side economics and the use of tax cuts (though Reagan also had to raise taxes on two occasions) as the reason for the growth, though Paul Volcker deserves significant credit as well for implementing policies that helped control inflation (though in fairness Reagan left Volcker in place and suffered the political fallout in the period it took for the policies to work). Reagan also oversaw significant deregulation of industry which supporters say encouraged investment and growth and detractors say paved the way for the housing bubble.
Defense Hawks in the Republican Party revere the role Reagan played in revitalizing American strength and ending the Cold War. Coming off the debacles in Vietnam and Iran, American confidence in the military was at an all time low in 1980. Reagan actively sought to turn around public opinion and make the US force more capable. This is probably the area of his most obvious success, as much of the way the public currently talks about the military began during the Reagan years. His effort started with the personal reading of MSG Benevidez's Medal of Honor (a presidential first) and continued with less symbolic moves like increasing military pay and massively increasing the defense budget. As early as May of 1981 Reagan began citing how a "new spirit was animating the land" and that the military was already rising in public regard. Investment in hardware to modernize the nuclear triad and increase the strength of conventional forces led to the B2 Bomber, F15, M1 Abrams, AH64, Bradley Fighting Vehicle and many other pieces of military hardware still in use. In 1985 the Joint Chiefs were uniform in the assessment that the military "was a far more capable force" than it was in 1980.
Reagan's greatest success was in helping the Cold War to end peaceably. While he is not the sole cause of this, and his role is hotly debated, he is incredibly important to the peaceful end of the conflict. Early in his presidency he recognized the cost Soviet spending on the military exerted on its economy. NSDD-32 talks about the need to force the Soviets to "bear the brunt of its economic shortcomings" that come about because of military spending. While the US did not spend the USSR into bankruptcy as commonly argued, the increased US spending and activity did led to a realization that the Soviets could not afford to compete and led to some retrenchment on their part. Local actors in Eastern Europe exploited these gaps fracturing the Iron Curtain. The relationship with Gorbachev is also extremely important, as the two understood one another very well. This led to success in negotiating arms deals, a decrease in the tension that defined Reagan's first term (and came close to war in 83), and also helped provide Gorbachev with the internal space he needed to make reforms. The modern Republican Party tends to idolize the Reagan principles of "peace through strength" and "crusade for freedom" that he employed in his relations with the Soviets.
With regards to failures, there are several large ones. The most notable is Iran-Contra, in which the Reagan NSC sold weapons to Iran for hostages and then used the profits to illegally fund the Contras. This led to a massive scandal which threatened to derail Reagan's second term. Ultimately his apology and the likelihood that he was unaware of the entire scope of the operation saved his presidency and the regard the American people held him in. Reagan's Middle East policy was also unsuccessful, most obviously in Lebanon where a US peacekeeping operation ended after a suicide attack against the Marine Barracks killed over 150. The War on Drugs also largely failed despite the escalation and militarization of it that occurred during the Reagan years. Domestically, the economic growth enjoyed by many did not extend to urban areas and his administration was incredibly tone-deaf on race, leading many to argue that Reagan at best didn't care about African-Americans and at worst was racist himself (I would characterize both positions as overly harsh).
Overall, Reagan was strong on the issues he cared most about (the economy and the Cold War) but his attention slipped on things he viewed as peripheral. His administration had significant success which outweighs the numerous failures and I'd argue that while he is not the saint the modern Republican Party paints him to be he is certainly not the demon portrayed by the left and was for the most part a good-to-great president.
There's a flaired user who is an expert on Reagan, /u/cubshound. I've generally enjoyed his or her posts on Reagan.
Additionally, here are some past threads that may be of interest:
https://www.reddit.com/r/AskHistorians/comments/30dvyd/why_is_ronald_reagan_so_praised_and_revered_by/ https://www.reddit.com/r/AskHistorians/comments/1ny2dh/why_are_there_such_polarizing_views_on_ronald/ https://www.reddit.com/r/AskHistorians/comments/1dyg9u/why_does_ronald_reagans_presidency_seem_to_be_one/
Finally, others can hopefully weigh in on your specific questions. I don't feel qualified to answer without indulging in my own political bias, so I shall abstain.