The book called Report: Disaster at Ionith in oblivion tells us what happened when the empire tried invading akavir during the second era. All 4 legions got wiped out at the end but would the invading forces had survived if the empire had send 10 legions instead of 4?

Well...
1. He could not send more than four legions at once. Two more legions were supposed to join in later however due to bad weather that was delayed. The disaster had already happened before anything could be done about it.
2. The extreme strain on supply that TEN legions would put on the area/mages (summoning food/water)/supply routes would propably mean that the army would have BIG problems supplying itself. Let us also remember that the supply was cut off due to bad weather for a year or so, the army sent however managed with what supplies they found/had in the area. Now, with ten legions being caught for a year on a limited amount of space... They would starve and the Tsaesci would win even easier.
3. Even disregarding all these logistical things... The Tsaesci pretty easily won over four legions with an unknown amount of their main force. If what drew the legions out was just a part of the main force... Then not even ten legions would stand a chance.


4. All the legions were NOT wiped out, only almost. The Fourteenth had some remnants left and two men from the Tenth survived.

Does it say anywhere in the lore about how much troops are in 1 legion?
And lets change it to 15 legions and the empire is able to magically get enough food and enough troops to form a gigantic invasion, would the imperials be able to go as far as encountering Ka Po' Tun and Kamals?

The problem with the invasion wasn't a lack of forces, it was a lack of resources and knowledge about the continent. More manpower would only have drained resources further, and wouldn't have prevented the trouble with weather, supply lines, or communication.

I don't believe the size of a legion is ever stated, but I'd say we can assume its equal to a Roman legion (about 6,000 men, according to wikipedia).

If the Empire was magically able to magically solve all its problems, it could magically conquer the whole world. But that's not how it worked out, and its impossible to know how big Akavir is or whether the empire could have gotten past the Tsaesci defenses.

If you read the book, you see that the issue was based on everything but numbers. Bad weather, possibly manipulated by magic, poor information, horrible logistics ect. were what contributed to the Empires defeat in Akivir, not the lack of manpower. So, had they sent ten legions, you would have had ten times the catastrophe and one seriously weakened Empire.

They also just waited, expecting the Tsaesci to surrender. Even after the messengers never returned.

While numbers can win wars, that way is costly in every sense of the word. Logistics are far more important in general, and logistics: the Empire didn't has it. That and the weather was so awful they say that it must have been sorcery.