cracks knuckles Well, seeing as I am a biochemist, perhaps I can take a crack at this. Though I would prefer some more specific questions, as this is rather broad. But I can go into a little on all of your questions.
Sequencing: We’ve got a huge number of techniques now, one of the first and most successful was Edman Degredation, which if you’re careful (and paired with other techniques such as proteolytic cleavage) can give you a sequence, though it’s rather tricky to do well. The most recent and powerful technique now is Tandem Mass Spectrometry, which can give you an unknown protein’s sequence in a day if you’re lucky and good. Basically it all revolves around slowly degrading the protein and seeing what falls off. Take that, pair it with powerful computer software, and you can figure out your sequence.
Along these lines, as @nikipedia mentioned, X-Ray and NMR spectroscopy can be used to determine angles and structure, as well as sequence, though sequence is very hard to determine just from those techniques alone. But NMR and X-Ray can give detailed structural information, especially related to bond angles and connectivity.
As to “They can’t just use a microscope to look at the molecules”, that’s actually not true. Scanning Electron Microscopy and more so Transmission Electron Microscopy can get very high resolution, with TEM getting down to almost single atom resolution. This has been used to determine the structure of proteins in ‘large’ molecules such as virus particles, though it has a lower resolution then X-Ray (~1 nm, as opposed to < .2 nm). Lastly, Atomic Force Microscopy has actually gotten down to single atom resolution, even further technically, though that involves a lot of signal averaging over lots of images.
However, a lot of knowledge on bond angles and things of that nature come from physics and physical chemistry, which informs biochemistry just as much as it does other branches of chemistry.
As to chemical reactions, there’s two things there. First, biochemistry owes a lot to the work of organic chemists (which i am decidedly not one), as most biological reactions fall under the umbrella of organic reactions, and physical organic chemistry has given us lots of tools to determine the exact pathway a reaction takes. However, in more complex biological systems, such as the photosynthetic pathway you mentioned, we use a combination of biochemical and genetic techniques, such as looking at where a protein appears in a genome and what’s around it (linked proteins tend to appear in a genome in close proximity, many times one right after another), then creating those proteins for study, along with organic and biochemical techniques (to determine how a single enzyme does it’s thing) like NMR and/or rapid kinetic spectroscopy.
And now I’m really late for work, so I should leave. But that’s a brief overview on many of the questions you bring up, if you want to get more specific into one I’ll be happy to give you more detail.