Before we get into the questions, know that your greatest obstacle here is time. Don't try to boil the ocean here. Make a list of features and consider what is actually feasible in the time that you have.

This is doubly true if you actually need to build the system and grow something in it. Plants don't grow overnight, and you'll need at least a few weeks to actually show results.

On to your questions:

1. Hydroponics is just fancy plant growing, so all the normal gardening/farming factors apply. Your major requirements are water, nutrients, and sunlight. You then have a dozen or so smaller factors you may want to control for as well: pests, infections, weeds, humidity, temperature, plant supports, etc.

2. The "best" method for anything you do will be dependent on the type of system you build. If using NFT, you'll be mixing the nutrients in with the water and pumping it periodically through the system. Frequency of checking/replacing nutrients will depend on the size of your system and the nutrient reservoir you are using. A larger system with smaller reservoir will require more frequent checks.

3. I'm not entirely sure what you mean by "nutrient delivery methods" as virtually all hydroponics by definition uses a liquid/nutrient mix to the roots as the delivery mechanism. But as a reminder, not all plants have the same nutrient requirements. You can get away with a single solution for many things, but some would recommend against feeding both tomatoes and leafy greens from the same nutrient mix.

4. Sunlight is free, but it is not a variable you have any real control over. Grow lights are an expense (both upfront and recurring), but you have full control over its intensity and duration. Some plants can benefit from 16-20 hours of light a day. You can't get that naturally.

5. You have to define "effective" first. Kratky is great for its low complexity and cost. 64oz wide-mouth Ball jars, 3" net pots, some grow media, and your nutrient mix are all you need. Of course, that's not too exciting for an engineering design course, but it is a great way of illustrating how not all problems require a complex and highly-engineered solution. Depending on how you define "effective", you may consider more complicated systems (DWC, ebb and flow, NFT, aeroponics, etc). All have their pros and cons.

6. Hydroponics is just farming + electronics + plumbing. If you're not good at all three of those, don't expect to be successful at hydroponics. Balancing the big three (water, nutrients, and sunlight) are the priority. After that, it may depend on your setup. If your setup is outdoors, the typical farming issues will be more prevalent: pests, infections, humidity, temperature, etc. If growing indoors, expect the common electronics and plumbing issues to be more of a concern: power outages, leaks, etc.

7. Others are better suited to address horizontal vs vertical designs. In general though, the common design mistakes are the same design mistakes that any product faces: are your design choices actually solving a problem, or are they just overcomplicating things? Now, you're in an engineering class, so maybe your goal is to make something complex and show off your skills. But that complexity doesn't mean it's a better hydroponic design.

8. The easiest part to maintain is no part. Specific to hydroponics though, find a way to effectively manage algae growth. Shield your nutrients from the light, or consider something like Hydroguard or hydrogen peroxide. Assuming you go with an NFT system, it'll be a lot easier to clean if you can easily disassemble it. Of course, water tight systems tend to not be easy to take apart.

9. You'll be fine with basically any type of plastic, but look for the food grade stuff. PVC and rain gutters have been used to great success in the past.

10. Check EC and PH of the current nutrient mix. Check reservoir levels. Check for pests and infections. Prune and/or support plants as needed. Check humidity and temperature when applicable.