I really need your answers as I have to deliver that flyer to the client tomorrow.
I commend you on venturing out. You might want to consider not taking on clients until you have at least a basic footing in file creation. Remember you don't just represent yourself when you "take on clients". You represent the industry as a whole. And bad word of mouth can reflect on any future designer someone comes across. I'm not stating you won't do a good job. Merely stating that these questions should really be knowledge you already have before any "client" is involved. Especially an overnight delivery client.
1) My document is 4.25 x 5.5 inch, so what should be the bleed?
For US design (assumed since you posted inches, not metric)... .25" or a quarter inch is a decent bleed. It generally encompasses any commercial printer you'll find. Some may ask for .125" (8th inch) some for .25". But there's really no such thing as "too much" bleed - in fact some more complex reproduction methods (gravure, flexo) may require much larger bleeds. So by using .25" you're generally covered for practically all offset printing. I tend to work in picas/points and I use 1p6 Picas or 18pts... which roughly translate to a quarter inch.
2) I placed the images in my flyer design, but I can see that those images [lost] their quality, how can I place them without losing quality,
By "place" I assume you are using the
Place command and inserting raster images into an Illustrator document. Raster images are imported at their set resolution. That means, if the raster image is only 72ppi, that is what Illustrator imports. Illustrator does not increase the native PPI of raster images ever. There are some caveats... such as effective PPI. See Here But in general, you need to ensure any raster image is already at the correct PPI (300) before placing it in Illustrator. And you should be aware that it is practically never correct to enlarge a raster image within Illustrator. You only degrade the image if you enlarge a raster image.
3) How to save the flyer as a PDF for print-ready with all the images and texts added? what are the options that must be changed?
Save as a PDF/X-1a File. Leave everything set at the defaults unless you've specifically been asked to change something. That's all you need to to generate a press-ready PDF.