1 ) Delta (for European options) is the mathematical derivative of the option price with respect to the price of the underlying (that's what the formula you wrote is doing). In mathematics, Derivatives are a fundamental tool of calculus. A derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). In the case of delta, it measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price.
Since a European call option gives the right to purchase the underlying at some predetermined price at some time in the future. there are only two possible extreme cases in between which the value of delta will fall:
- Delta = 0: the price of the derivative does not change if the underlying changes: this is the case for far out the money options - e.g. the right to buy something in 2 days for the price of 1000 that costs only 1 at the moment (it is worthless). This corresponds to the very left hand side of the graphic below.
- Delta = 1: the price of the derivative changes one for one with the price of the underlying e.g. the right to buy something in 2 days for the price of 1 that costs 1000 at the moment (it is certain that the right will be used, hence the movement is identical to owning the underlying outright). The very right hand side of the graphic below.
Geometrically, it is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. In terms of option pricing, you can plot the price of the option with respect to different values of the underlying, while keeping all else constant. The delta is the slope at any point you look at (at any given value of spot). It helps to look at this in a simple graph (done in Julia, code can be found here, how to make it interactive can be seen here, albeit for theta).
Side remark, that is very much simplified, because in reality you can have Spot, Forward, Spot premium adjusted, forward premium adjusted Delta for example. Non Vanilla options are anyhow different and delta can be very different from 0-1.
2 ) The asset trend has nothing to do with option pricing. Delta hedging is done so that the portfolio value remains unchanged when small changes occur in the value of the underlying security. You do not want to bet or guess the direction of an underlying. That is gambling. Naturally, this hedge is only short lived and works only in the near proximity to the current market conditions (a linear approximation to a curve can only do so much) . That's why market makers frequently need to rebalance if they want to hedge their exposure.
Edit: I know you ment the trend in the underlying security but as I wrote, this trend is entrirely unimportant for pricing options. In a nutshell, option pricing is entirely based on "no arbitrage" and "replication / hedging" arguments within a risk neutral framework. As a result, the derivative price does not depend on the drift / trend of the underlying.