I think the mere length of a sentence is only a small part of what determines how readily it's understood.
Vocabulary, syntax, and, most important, contemporary grammaticality are also important.
If I write a long sentence that has a simple structure, uses very common words and adheres completely to modern grammar, it may be easier to understand than a much shorter sentence with archaic or obscure words, poetic semantic shifts and multiple negatives.
When you look at old texts, apart from the lengths of the sentences, you will also stumble upon mundane things like spelling and vocabulary, as proven by the sentence quoted by Claudiu in the topic you linked to:
In this season the legat vpon his partye, and the kynge of Romayns vpon ye other partie, for allyaunce that was atwene hym and ye erle of Glouceter, laboured so to the kynge that a reformacon of peas was spoken of; durynge whiche treaty, the souldyourrs lyinge in Southwerke made many robboryes in Southerey and other places, and rowed ouer to Westmynster, and spoyled there the kynges palays, and deuoured his wyne, and breke the glasse of the wyndowes, and all other necessaryes to that palayes they distroyed and wasted; and somtymes came in lykewyse into London, and robbed there also.
It is indeed unlikely that you will read this sentence once and understand it fully, because your reading gets interrupted by many things apart from length:
Spelling the u as v (and vice versa!) is something you can get used to, but it is bound to make you stumble at first.
Words like kynge or hym are bound to interrupt the flow of reading unless you manage to read phonetically at all times.
(Mis)spellings like ye, pronounced the (!), are another reason to break the flow.
Outdated grammatical constructs like came in into London or robbed there also make it even harder to keep on reading.
Now, all these points are moot for contemporary readers, so for them, the sentence will have been easier to read. Actually, even though the sentence is not a short one, I believe that once we eradicate the earlier points, it is quite a simple and straightforward sentence.
I don't think you can make a fair comparison on the difficulties in understanding sentences based on their length if you do not keep the other factors constant. So to compare sentences form times when few people could read, and the ones that did read were usually educated far beyond the point of simple literacy, to modern, to-the-point writing, possibly aimed at an audience that struggles with a newspaper headline if it has a long word in it, is hardly a worthwhile exercise.
To get to the point, let's look at your three questions:
1. Are these problems just my own, or do other contemporary readers suffer as well?
It is unclear whether you refer to old texts or newer ones. For old texts, I am sure you are not the only one. Not just because of sentence length, but simply because of the dated (use of) language, I believe many modern readers experience, at least at first, trouble in understanding them.
For modern texts, I believe the length is also not the only problem. Often we find long sentences in specific, specialized areas of language usage, like in legalese. Not just the length, but the almost unavoidable presence of specialized jargon ensures that many people who are not used to or well versed in the specific jargon will struggle to understand such texts. Ask people to explain details of their labour or mortgage contracts (or general terms and conditions for the latest software they bought!), and you may be surprised at the level of understanding many people have of documents they readily sign.
2. If so, what are some strategies or tips on determining their meaning more efficiently?
First, deal with the other, simpler problems. Get used to, and understand, the jargon used, or the contemporary expressions, vocabulary and semantics of the text. With older text, it often helps to simply practice reading - as an example for Shakespearian English, wrestle yourself through some plays, and at some point you will be so used to the language that the rest reads almost as easily as a Sunday sports section.
For (professional) jargon, you might need to dig a bit deeper and use (specialized) dictionaries, reference books, and if possible, annotated versions of the texts you want to read.
Once you are used to and familiar with the vocabulary, grammar, spelling and semantics of the text, you will be able to focus on understanding the flow of a sentence while reading it uninterrupted. This will greatly enhance your understanding, since you will be able to analyse the complete sentence structure, no matter the length, without distractions.
3. What about readers bygone? Did they apprehend and read the denser sentences more quickly?
I am sure that in the case of some (legal) texts, readers of yore had similar struggle you and I experience when we try to read the updated terms and conditions for some on-line service. On the other hand, in many cases, novels and stories were aimed at people who, if they had learned to read, were likely to also have had more education beyond those basics (you were either well educated or illiterate, to put it in a grossly simplified way). So things that might trip us up, like historical references or bits and pieces of French or Latin, were likely readily understood.
And of course, those bygone readers did not have to put up with archaic spelling, grammar and semantics like we do when we are reading their texts. So while they may have had to read some sentences twice or thrice to understand them, it is unlikely they were taken aback as much as we are today by the language they encountered.