Because I apparently can't plug the Uni-Wiki enough, their page on Research explains most of what you are looking for:
Skills Required
- Laboratory Operation I
- Research I (5% bonus to blueprint manufacturing time research per skill level)
- Science 1 (5% Bonus to blueprint copying speed per level)
- Metallurgy 1 (5% Bonus to material efficiency research speed per skill level)
- you should have Production Efficiency trained to level 5 before you even start producing.
Helpful Skills
- Advanced Laboratory Operation (+1 lab slot per level)
- Cybernetics 5 (allows to plug in special implants that speeds up the research process)
Implants
- The Beancounter Hardwiring series
You can reduce this waste by researching the Material Level (ME) of the BPO. You can see the design waste on the BPO under Wastage Factor. This is always 10% (0.1) except for some rare items that have a 5% (0.05) Wastage Factor. Drones used to have a 5% wastage factor, but apparently they don't anymore. By researching the BPO we can reduce the design waste using the following formula: 1 + Base Wastage / (1 + ME level).
Using the above formula on a BPO with 10% base wastage, we can calculate our production waste (assuming PE skill at level 5) to be:
ME0 - 10%
ME1 - 5%
ME2 - 3.3%
ME3 - 2.5%
ME10 - 0.9%
ME50 - 0.2%
As you can see, training your ME level from 0 to 10 reduces the waste from 10% to 0.91%, thus reducing the waste by over 9% in simply 10 levels, whereas training a further 40 levels (11-50) only takes off 0.7% (from 0.9% to 0.2%). The following graph can better illustrate the effect of training ME to higher levels.
A general rule of thumb used by the producers is that 30 days of research is usually sufficient.
The Productivity Level of a blueprint, as mentioned above, affects the manufacturing time for each batch made with the blueprint. Specifically, it reduces the manufacturing time by (Productivity Level) / ( (5 + 5 * Productivity Level) ) which results in the following progression:
PL 1: 1/10 reduction in time
PL 2: 2/15 reduction
PL 3: 3/20 reduction
PL 4: 4/25 reduction
PL 5: 5/30 reduction
...and so on.
For example, suppose that manufacturing a batch of something normally takes 60 minutes to manufacture at Productivity Level 0. At Productivity Level 5, this is reduced by 5/30, or 10 minutes, thus the manufacturing time for one batch is 50 minutes.
Increasing the Productivity Level of a blueprint is usually not as important as increasing the Material Level of a blueprint. It might be important in cases where the blueprint is being used to manufacture almost continuously or in cases where the manufacturing time for a single batch is extremely long.
For some, quite unintuitive, reason - these seem to be stored over at the POS, rather than being delivered to station where the process was started like the original blueprint. To access them you will need the corporation rights to access the storage of the mobile lab you used to copy the blueprint and then they will be in the destination hangar you originally selected;
Best Answer
Time research reduces the time production runs take. I think level 20 is generally considered the sweet spot in terms of usability.
Production efficiency reduces the amount of waste - and thus lets you produce the same amount of stuff for slightly less resources/components/etc. As with Time research, there are diminishing returns on higher levels. I think 20 is regarded as the sweet spot, but check the forums for specifics per blueprint.
Invention is used to turn a tech 1 blueprint copy into a possible tech2 blueprint copy. This requires a lot of skills plus datacores. A succinct explanation of how the entire concept works can be found here.
Reverse engineering is used with relics and artifacts found in archaeology sites, and can be used to develop specially modified hardware, as well as being crucial for tech3 ships.
The reason publicly available slots tend to be full is because, well, they're publicly available -- and there are a LOT of pilots. Best bet is to join a player corp with their own stations and a research division.
EDIT: This link on the forums has a lot of good articles to sort out what's what with inventions and industry in general, and this link in particular has pretty much everything you want to know about research in specific