Reddit mentions of Apollo Horticulture 4" 6" 8" Inch Inline Fan with Built in Variable Speed Controller - Choose Your Size (4" Inch)

I don't know very much about those booster fans. I have never used one but i have heard that they are not very powerful. I would try to get something like this one. The carbon filter you linked looks good though.

You need more than 36" vertical space. Your fan + filter will take up 10", pots another 10", and you will need to keep the light 12"+ away from the plants. This tent is close in footprint, but with a much more manageable height. Tons of good tents out there though. Just make sure you have at least 5' if you are flowering. 6 or 7 is better.

Can't speak to that particular Mars light. If you only have $150US/100GBP to spend, I'd look into more options. Mars300 for a first grow is an idea. A better idea, if you have the time/inclination, is to build a light. You can get diode strips (Bridgelux EB, Samsum Series H, etc) + 100w driver for the same price, and it'll put out more useful light with less heat. If that sounds like something you want to do, search a bit on this subreddit, and post questions. Lots of builders here.

As for fans, Active Air is a solid brand. Just don't buy anything that look like this--which is a duct fan, not an inline or this--which is an inline that will work for 3 months then break.

Probably not. What kind of fan is it? Moving air through a filter is very difficult unless the fan is designed for that purpose.

This would probably work, but this would definitely not. Even though it has more CFM. I made this mistake a while ago, and wasted money I could have spent elsewhere.

I assumed you had a good inline fan for exhaust, as they’re necessary to help keep humidity down in late flower (or you risk mold). You can get away with passive intake (no fan, just a hole), but you’ll need a good exhaust fan.

You can do it on Amazon for ~$80, and the $45 fan should last you many grows. I recommend this 4" inline fan since it comes with a speed controller, and something like this filter, since it's cheap but will actually get rid of smells for a couple grows.

Here's a 4 inch fan with a built in variable adjustable knob from Amazon for less then $46 with free shipping.

Apollo Horticulture 4” Inch 190 CFM Inline Duct Fan with Built In Variable Speed Controller https://www.amazon.com/dp/B019INGAO4/ref=cm_sw_r_cp_apa_OYfUAb51BZ9S2

So that isn't an inline fan, that is known as a duct booster. It's just meant to help keep air moving through ducts. You'll need a real inline fan with speed controller. http://www.amazon.com/Apollo-Horticulture-Inline-Variable-Controller/dp/B019INGAO4/ref=sr_1_3?ie=UTF8&qid=1464883260&sr=8-3&keywords=4+inch+inline+fan