Which Firewall of Fortigate help me in improving the network.

If this is a casting issue, that would be an issue with WiFi coverage.  Not the firewall... What does your RF environment look like?  Have you done a wireless survey?  Is your wireless network properly designed for your client coverage and capacity? 

Where are you getting the information on 200 users?  Is that the SSL VPN number?  FortiGate firewalls are sized on total throughput, not users.  I would work with your preferred Fortinet partner or your account SE on a sizing discussion.

No, you should not add a Dream Machine and a FortiGate.  The Dream Machine would replace the FortiGate and is not needed.  FortiGate is a FAR better firewall than the Dream Machine.

Alternatives for FortiGate?  Why?  Palo Alto, Cisco, CheckPoint are some very common competitors.

Hi @szcad 

You need a new sizing in order to find which FortiGate model is best for you (and probably for WiFi as well). As suggested by @adambomb1219 you should ask your Fortinet account manager or you Fortinet partner for such sizing.

"Will Adding Ubiquity Dream machine pro along with the fortigate Firewall help in improving the network."
Most of the time it is better to buy the same firewall twice if you are going to buy two anyways.
If you buy two FortiGates you can put them in either Active-Active or Active-Passive HA. 

Whew, you got a lot going on and will restate some of what others have mentioned, but also give some opinions from my perspective (and making some assumptions from lack of further details), also coming from a school district. I would recommend a consultant to come in and riddle out your network infrastructure issues with WiFi.

Devices getting disconnected from WiFi have nothing to do with the size of your firewall based on your topology statement (as it's just an edge firewall). As others mentioned, the firewall sizing is more from a throughput requirement than a user requirement. That being said, your 150Mb Internet circuit is woefully undersized and I bet you are maxing that out daily (you have staff as well to take into consideration as well). Based on my typical needs of throughput (especially during times of state testing), my guess is 1Gb would be more appropriate for your size district.

Hopefully, you can address this via e-rate funding which would pay for a majority of that cost. With your circuit maxed (again, making an assumption), you will get a lot of dirty traffic, packet retries, etc... So for example, you mentioned screen cast disconnects, this could be the result of your wifi issues, but this could also be the result of your Internet circuit saturation as some platforms use a cloud server for their screen share to connect to (eg... Promethean), so if that cloud server cannot be reached during periods of congestion, then you would have screen sharing issues.

The 100E model you have is rated at 250Mb Threat Protection, so it can handle your current Internet circuit bandwidth; but if you are able to upgrade your circuit, combined with  evolving off the older E series of firewall, I'd probably look at the 100F (1Gb Threat Protection) or 120G (2.8Gb Threat Protection) series...again my opinion.