Does warmer temp result in rapid fermentation?

On my current batch I am using a cooler full of water and frozen bottles to try and keep the temperature down since my house can get up to 75 at times. I've noticed a few differences this time around:

1. Fermentation took longer to start
2. Fermentation wasn't as fast
3. Blow off wasn't as dramatic (barely any)
4. After 6 days still have high rate of bubbles

I'm wondering if this is the result of the lower temps. With my other 3 batches (no cooler) the fermentation was fast, required a blow off hose, and after maybe 4 days slowed to very few bubbles per minute.

The other variable is this was the first batch I did with liquid yeast and a starter so that might be it also.

I'm not concerned about it yet, but I am curious to know what "normal" is since this is different than my other batches.

short and simple answer: yes, lower temps will slow down fermentation.

Why is slower fermentation a good thing?

Welcome to HBT!
Slower fermentation isn't necessarily better but lower temperatures are very good.

A fast, warmer than normal ferment can cause your yeast to create more off-flavor producing byproducts. More esters, more hot alcohol flavors.

Ferment temp can be just another thing to manipulate to get the beer you want in the end. If you make certain belgian styles, like a saison, you'll want to ferment warmer to get those flavors. If you are making something like a blonde or a cream ale, you'll want to keep a tighter grip on your temps and keep them low since off flavors will stick out like a sore thumb.

Having the wrong temp isn't going to automatically ruin your beer, but if you can get tight control over your fermentation temps you will have more control over the finished product.

What's the best way to get a grip on fermentation temps? Modify a fridge? Room temperature control?

What's the cheapest and most efficient way to get the job done?

cheapest way is with what we call a Swamp Cooler.
Basically, get a large plastic beverage tub, put carboy or bucket in it. full tub with water, place wet shirt over carboy/bucket. as the water evaporates from the shirt, it will wick water from the tub, this cools the beer. if you need to drop the temp more drastically, you can add frozen bottles of water to the tub. by rotating bottles from freezer to tub, you can lower the temp even more. i have been able to get fermentation down to 65 in an 85 degree room this way.

Cheapest would be the tub of water and ice bottles.

I think the best balance between cost and ease of use would be to get a fridge or chest freezer off craigslist for as cheap as you can and then get an external temp controller. Then you just set the temp, load your carboy(s) and walk away. Plus, you can cold crash to clear up your beer at the end.

I'm still at the tub and ice bottles stage, but once we get into our house and aren't in our little apartment anymore, I'll be working my way into a chest freezer fermentation chamber.

I've kind of hit a sticking point with cash. The central AC stopped working in my home, so all extra cash is going to replace that. In the meantime, I'm checking CG for a free fridge/freezer and will probably go the swamp cooler route...

Thanks for the advice!