As you have no doubt realized, the base-て conjugation of verbs is used for more than just imperative commands. I'm sure you've seen many cases where you have a grammar structured as with the
～て + _____ structure.
What you are seeing is something that is not uncommon. What is happening is that the speaker/writer is using the masu form of a verb to maintain (or increase) politeness, but they are also using some other grammar as well. Sometimes, it is laid on a little thick or overdone.
You typically won't find this in Japanese text books, but it is still something that is done commonly by native speakers. (See this source.)
Beginning Japanese speakers should not need to worry about using masu in this manner, as it is sufficient to have the final verb of the sentence in the masu form.
Most often, you will see this used in sentence conjunctions, or in situations that leave the sentence open-ended. That being said, it is not very common, but I've also heard ～ましても (i.e. ～て + も), and other grammatical structures as well.
Be aware that not all structures using the base-て conjugation will work seamlessly with masu. Some grammars won't work with this structure at all, as you can see in this answer. It's something that you will likely have to get a feel for over time.
Is this construction in any way related to はじめまして?
This is something that has become a set phrase with a specific meaning. I did some digging on this site, and found an excellent post by @chocolate that answers your question quite succinctly. I'll quote part of it below, but I recommend reading the full post.
To break it down, はじめまして is the て form of はじめます, and はじめます is the polite form of はじめる.
はじめまして is a fixed expression.