Does mentioning “Cheap” in a service domain name leave a bad impression?

Think about how other low cost offerings emphasize the discount level. Fiverr, for one. Cheap is a ticket to failure. For many, cheap means "easily breakable, unreliable junk." Although Google guards the information with its life, I also suspect that Google devalues the word cheap from an SEO perspective. Even the most successful three domains with the word "cheap" in them have a fairly low page ranks - between 4 and 6 - and receive a less than impressive amount of visits per month (estimated with Compete.com).

I can help you choose a more effective domain name that communicates the same message and is more relevant to search volume, but I need more info on your business.

Traditionally yes!
In our digital-era, consumers are more aware than ever, and so cheap tends to give us a red flag that is cheaper due to something besides the provider being nicer and thus charge less. People don't want to know is cheap but what they/we care about is whether we get good value for our money, that the quality is superior to what we pay... If you are talking about software refer to your quicker turnaround and thus that causing fees to drop, if is a product focus on quality and reliability made affordable, etc.
there are many ways to approach "being cheaper"

Best of luck!
Not to self promote but if you you wish me a call!
I don't have much feedback here but I've been doing this for a while :)

Yes, it does sounds cheap to actually use the word "Cheap" in your services domain. There could be better ways to communicate the same, some of them being "Value for Money", "Cost Benefit", " Cost Efficient", "Cost Advantage" et al.

Having cheap in the domain name is generally not a good idea- especially because rebranding means changing domains if the strategy doesn't work. That said, marketing yourself as the "cheapest" service is fine. Go low end with the design and the price.

The best way to judge is to A/B test a landing page and put some money behind some Facebook and Google advertising and see which gives you better conversion rates.

There are cases where branding with the word "cheap" might be fine, but they're outnumbered by cases where that isn't the best idea.

If you offer 1 product that is literally identical to what the other people are offering and your entire business model is based around undercutting their prices, then "cheap" might be perfectly appropriate.

For example, if you were booking hotels or flights – the very same hotels and flights as the other guys – but at cheaper prices, then I'd see "cheap" as an important part of your identity.

Without knowing what your service is, it's impossible to give specific advice. Happy to chat for 15 minutes about your branding / domain name options. That's what I do full time.

greatqualitycheap.com has a certain ring to it!

Do you ultimately want to be perceived as "Walmart" or "Nordstrom"? If what you're emphasizing is low price, then that's who you'll attract and that's what you'll be competing on.

Raising prices in the future would be very difficult as well.

It would be more satisfying, (and your margins will be higher) if you stress the value they would achieve by using the service.