For printing a poster, should I go for 8 or 16-Bit mode?
16-bit color is usually overkill for most any project. Professional photographers will often use it for the flexibility it provides when editing RAW imagery, but beyond that, it's not usually something you'd need to deal with.
For putting my poster on web, what should be the ppi because I think 300 is just too much?
On the web, PPI is completely irrelevant. PPI settings in PhotoShop are only relevant when a) you are printing and b) the software you are printing from recognizes PhotoShop's PPI setting and automatically sizes your image to match. (This means that most of the time, the PPI setting in PhotoShop isn't really relevant outside of PhotoShop.)
For the web, ignore PPI. All that matters is the pixel dimensions. If you want to display your image 800 pixels wide, make it 800 pixels wide.
Also should I go for 8 or 16-Bit mode?
I'm not aware of any file format on the web that supports 16-bit color.
As you know that my computer lagged while editing, so if I'm designing an A3 size poster for printing, can I lower the quality(e.g. 100ppi, 8-bit) just for the sake of ease of making it/prevent lagging and once the poster is complete, switch it to optimum quality as before?
You can but...
Will it effect anything provided I'm using high-res images?
Yes, you will essentially be working with a 100ppi image. While increasing resolution in PhotoShop can improve the image quality slightly, it can't magically turn low-res imagery into hi-res images very well. So you typically don't ever want to increase the resolution of your image as the gains aren't there.
All that said: Note that posters rarely need to be 300ppi. The 300ppi 'standard' typically refers to quality print work done for hand-held reading. Typical example being decent quality magazines.
A poster is rarely read at the same distance as a magazine. Most people will view a poster from 3-10' away. At that range, 300ppi is overkill. You likely can get away with 150ppi in a lot of cases (if not even lower).
Finally, consider using a vector-based tool in the future such as InDesign or Adobe Illustrator. This is often the best of both worlds. For instance, you may import a background photo at 150ppi, but then set your type in Illustrator on top of it, giving you the maximum resolution for your type that the printer supports.