Are there any measures to ensure "academic honesty"?
Let me throw in a few points in addition to @Peter Janssons great answer:

There are "tools" that help with some of your concerns


  that the reviewers do not lose their objectivity, for instance when looking at a manuscript of a competitor



For one thing, most journals I know not only ask for suggested reviewers, but you can also ask that particular persons are not asked to review your paper.
The journals I know have at least 2, usually 3 reviews. So a firm "reject" means that several reviewers did agree that the paper is bad. While one may have a biased opinion, with 3 rejects it is maybe time to stop and think whether the paper really is good or not.
Bias in peer-review is one thing, but variance is another. And, from my experience, variance is high. IIRC there are studies about marks on texts by different school teachers which were found to vary between good and barely passed for one and the same text. My guess is that peer review is similar. 
Personally, I'd prefer if peer-reviews were done openly (both sides named, some journals do, though none in my field) as opposed to one side more-or-less blind.  



  that grants/prizes/titles are actually given to the better projects/people from the perspective of the greater good, and not based on how well-connected the applicants are? (after all such committees need to have or be composed of other researchers, who else is going to be able to judge the impact and importance of cutting edge research than other researchers?)


I think this is fundamentally impossible, because that would imply that it is possible to know which project will be a good project beforehand*. This may be possible for very applied projects, but those would typically be industry projects, not basic research.
The "interesting" projects always imply a risk that the thing doesn't work out, no solution is found. Otherwise no research would be needed on the topic. 

Sometimes, the huge importance of projects is clear only years (or even decades) after they are finished.

* On the other hand, I think it is possible to find out in advance that a project is poorly designed, so it should not be funded. 


  that there's no collusion between high-profile PIs and editors of "high-impact" journals?


Of course that can happen. Corruption exists, and I don't think science is fundamentally different from other fields of human professions.
(see below for my thoughts on how to deal with that)


  that academics (at varying stages of their career) do not consider/prioritize "pushing up" the numbers (e.g. "h-index" or "impact factor" etc) when they set out with their research projects?



In a way we cannot, and IMHO it has to be expected that scientists are intellectually quite able to understand how to optimize a given measure. Thus, the assumption that e.g. bibliometric measures do not influence the measured system will not hold. 
Whether they actually do it, is another question. 
There is one very important point: the papers are not kept secretly, you can read them. As a scientist, you can, should and do judge the quality of the papers. IMHO peer review does not allow you to switch off your brain when reading paper - I think peer review is meant to be a sieve that gets the "failed" papers out of the system - while what I want to read are not "barely passed" but "excellent" papers. However, there is a tradeoff, and if you push up the rigour of peer review you'll inevitably throw out also good papers, or papers whose importance will become apparent only later on. That is, too strict peer-review may pose a restriction on the publishing system that disfavors really new ideas, and allows to pass only predictable results.
However, you won't be able to help realizing who publishes by salami-slicing and who publishes lots of studies with e.g. very low numbers of patients, poorly designed experiments, and OTOH, which groups take the effort to get meaningful numbers of patients, publish on relevant controls, do their homework validating their findings and so on. 
Also the authors are of a paper are named clearly. And taking together your judgemen of the quality of papers and maybe some citation network tool, you can get an idea  who optimizes e.g. citations. 
Attending conferences and workshops and talking to people you'll get to know people personally so you can judge their character. In addition, you'll hear a fair amount not only of "rumours" and "stories" which also tell about people.  


All in all, I think this works as "real life" does: how do you know a company is honest you consider dealing with? You put some advance trust in them, and you use your judgment, including what you hear by your peers about them: e.g. were they recommended by someone you consider reliable.