Effort setting game - no idea where to start
This problem is similar to first-price (sealed-bid) auctions with independent private values for the buyers. You can read about those for exaxmple here.

Your first question was already ansewered in the comments: given others' efforts the probability of getting the reward is  or 1. Let us assume that $e_1 \geq e_2 \geq \dots$ are the other players' efforts and $e$ is own effort. Then
$$ \Pr[\text{getting }R | e, e_e, e_2, \dots] = \begin{cases} 1 &amp; \text{if } e &gt; e_3 \\ 0 &amp; \text{if } e &lt; e_3 \\ \dots &amp; \text{otherwise} \end{cases} $$
What happens in case of ties is not exactly specified in the question, but as $\theta$ has a continuous distribution, if $e(\theta)$ is strictly monotone 
edge cases happen with 0 probability. Based on this, you can also calculate what happens if you do not condition on other players' strategies (you should probably look into order statistics).

Now the player want to maximize the following:
$$ EU(e) = \Pr[\text{getting }R | e]\times R - c(\theta, e) $$


You can start by assuming that players follow pure strategies, so $e_i = e(\theta_i)$.
Also assume that effort is strictly monotone decreasing in $\theta$.
And last, assume that players have symmetric strategies.


Now you have to write down the expected utility defined above in terms of integrals. As of assumption 2. (and the convexity of $c$) you can differentiate it with respect to $e$ to get a (sufficient) first order condition. Then using assumption 3. and luck will get you a solvable differential equation, and hopefully you can find a closed form for $e(\theta)$. That is going to be your equilibrium strategy.

Or to be precise, one of the equilibrium strategies, as we have only looked at a subset (pure symmetric strategies). But finding others (or proving that none exist) is much more challenging.

I hope this is enough to start. If you get stuck, I will gladly expand this answer with some concrete formulas.

Disclaimer: I have not solved this problem, but as it is remarkably similar to the previously mentioned auction problems, this method should work.

There might be another way using the revelation principle. You could come up with a (second price auction-like) game where reporting the truthful $e$ is the dominant strategy, and has the same expected payoffs. Then you could relatively simply calculate the expected payoffs, and based on that, getting the strategies might be quite easy. I am not sure this method works either, but if you are interested I can look into it.