1. <em>My family is immune to</em> the <em>flu as everyone got a flu shot.</em>

"The" is there to modify the noun and make it more specific. Example: "We read the book." 

2. <em>Nobody can</em> induce <em>me to change my mind. Regardless of what people say I am going a graduate school in the UK.</em>

> Nobody can induce doubt in my mind. Regardless of what people say, I am going to a graduate school in the UK.

Induce here isn't used as a verb right since you're not exactly indicating what they are inducing in your mind.

<em>3. Food in convenience store is bad for our health, because a lot of synthetic additives are used.</em>

> Food in convenience stores are bad for our health because a lot of synthetic additives are used. ✓

Food is a lot of things - it's not just one food that's in a convienence store, so you'll have to put the "are" to indicate multiple things and the "-s" at the end of stores because you are making a broad statement about food in convienence stores. I also find that the sentence flows better without the comma -- the reason the comma isn't necessary is because you are connecting only two sentences together.

<em>4. While many western people often discuss about their thought, Japanese do not like to intrude our opinion to others.</em>

While many Western people discuss their thoughts often, Japanese people do not like to intrude upon others opinions.  ✓

Western is capitalized because it's indicating a selection of people -- just like Japanese. 

<em>often discuss about their thought --> discuss their thoughts often</em> 

The opinions are multiple THOUGHTS -- not just one thought. It's important to learn about when there's a lot of things implied in a word and when it's only one, for future reference. Often discuss their thoughts is as correct as discuss their thoughts often, but in my opinion the latter works a lot better in terms of ease to read.

Japanese do not like to intrude our opinion to others. -- > Japanese people do not like to intrude upon others opinions.

Just like how you said Western people, you should say Japanese people too. 

Intrude our opinion to others isn't right because intrude isn't actively projecting onto others. My revision is more suited because "intrude" is more like saying someone makes you feel unwelcome, rather than yelling about your opinion. Again, the "-s" is at the end of opinion and other because there are more than one.

<em>5. My maneger manager is good at manipulating workers.</em> ✓

<em>6. We need to obtain a patent to sell our products.</em>  ✓

<em>7. Plenty of scientists probed about around the meteor.</em>

It's not incorrect, but it flows kind of weird. "Scientists are probing about the meteor" would fit the word "about" better.

<em>8. Dr. William provoked a scientist by desagreeing disagreeing with his research.</em>

You need the "with" to link the verb with the noun. 

<em>9. A placebo remedy is not very common in Japan.  </em>✓

<em>10. I want to surpass people in my university.</em>  ✓

There are probably some errors in my corrections since I'm tired, but this should outline the gist of what mistakes you made and I hope I helped a little! Good luck ^^