Ever since GoDaddy's DNS outage I definitely don't trust name servers.
I just created a domain name and next step is setting up name servers. I'm thinking of using Route 53, CloudFlare DNS, and another provider. This means I'd probably end up with about 13 name servers attached to my domain name.
Should I limit that amount? Main concern is adding a tiny bit of latency each time someone does a lookup on my domain name. Does the lookup download the list of 13 name servers?
There is such a thing as "too many nameservers", but not for the reasons you're concerned with. The others have covered the irrelevance of latency concerns and I won't beat that horse to death.
The real problem with adding too many nameservers is authority bloat. A reply to a SOA record request should not exceed 512 bytes if at all possible, including both the authority and additional sections. TCP and EDNS can be used to overcome the 512 byte barrier in most cases, but some nameserver software still behaves poorly when the authority+additional information can't fit within a standard 512 byte response. (mostly in regards to zone transfers or DNS forwarders)
Taking the above into consideration, the answer on how many DNS servers is too many still boils down to "it depends". You should plan on your DNS servers supporting IPv6, so that's going to add overhead. (A+AAAA records)
For reference, here is what Yahoo's SOA reply looks like at this point in time:
As you can see above, six NS records along with their corresponding A and AAAA addresses are already weighing in at 411 bytes. If I were to add so much as six characters to the base domain (6 * 19 = 144, 411 + 144 = 555) I'd already be breaching 512 on mandatory records alone, nevermind optional bloat this usually adds to the right hand side of the SOA record.
Most companies don't use more than four NS records. Six is probably too much. There aren't many good reasons for extending into this number so long as you're following the usual guidelines, or using a well-respected DNS provider.
Let's be clear. Ever since the outage of a low cost low end provider of whatever you do not trust a technology that is proven. Why? You think everyone is as incompetent as GoDaddy? Never had a DNS issue in years.
No. You have no clue how many name servers. What you see as one server can be a cluster using the same IP address. You only have a minimal amount.
Seriously, you totally overestimate the issue. If you use CloudFlare than this is good enough - especially if your content is behind CloudFlare.
Which happens exactly how often? Remember - DNS answers are cached, possibly on multiple levels. I.e. I would bet most end users use their provider's DNS. You REALLY overthink the estimate of that 1ms (max) in the whole larger amount of work it takes to get to the host name and then make something with it.
Overengineering showing a paranoia mostly built on not knowing technology AND Professional providers. Get rid of that.
I personally use CloudFlare for everything and I am happy with ONLY them for my DNS.
RFC2182 section 5 recommends at least 3 nameservers, and preferably no more than 7.