Why is 开 written within a 门 in this sign (without being written 開)?
BabelStone wrote a Proposal to Encode Obsolete Simplified Chinese Characters which also contains a lot of history and other important information.
Under the heading 2.2 we find:

Table of First Batch Simplified Characters
The table below lists all 324 first batch simplified characters in the order that they occur in the published table of simplifications. The reference number is prefixed “DJ” for Dìyīpī Jiǎntǐzì. Mappings to the corresponding standard simplified and traditional character forms are given in the columns marked “Simp.” and “Trad.” respectively. Characters that are proposed for encoding are given a Unicode number of U+XXXXX and printed in red. Unicode points marked with an asterisk are for CJK-D characters, and as of writing are still provisional.

Here we find:


DJ‑89  U+2B52D*      开   開   kāi

Background about these simplifications also on the BabelStone proposal explain under 2.1

First Batch Simplified Characters (1935)
Background
In January 1934 at the 29th meeting of the Preparatory Committee for the Unification of the National Language (Guóyǔ Tǒngyī Chóubèi Wěiyuánhuì 國語統一籌備委員會) of the Ministry of Education, Qián Xuántóng 錢玄同 (1887–1939) submitted a draft table of simplified characters (Sōucǎi gùyǒu ér jiào shìyòng de jiǎntǐzì àn 搜採固有而較適用的簡體字案). These were approved by the committee, and Qián Xuántóng was delegated to edit a table of simplified characters for publication. In August 1935 the Ministry of Eductaion published a first batch of 324 simplified characters (see Appendices A and F), with the intention of publishing a series of further batches of simplified characters. However, in February 1936 the Ministry of Education issued a directive halting the simplification program, and no further batches of simplified characters were issued. Although this first batch of simplified characters was not widely used, it was historically very important as it represented the first centralized attempt to promote the use of simplified characters in China, and was a precursor to the simplification movement of the 1950s and 1960s in the People’s Republic of China.
Fig.1 Jīngběn Tōngsú Xiǎoshuō 11:2a

DJ‑228:  (錢)
The simplifications in this table were based on commonly used vulgar form characters (e.g. 宝 for 寶), ancient forms of characters (e.g. 无 for 無) and cursive form characters (e.g. 为 for 為). Most of the 324 first batch simplifications are identical to modern standard simplified characters or Japanese simplified characters, but some are different to a greater or lesser extent; and those characters that are not currently encoded need to be encoded in order for scholars to discuss this pioneering episode in the history of Chinese character simplification. Moreover, some of the currently unencoded “first batch” characters are commonly found in early printed texts (see Fig.1), so it would also be useful to encode these characters for scholars of such texts.

Most likely the building was founded during Aug '35 - Feb '36 by some forward thinking intellectuals who were keeping up with the times.
As in your picture we can see that 學 is also simplified but 現 is not.
We can find 學 as the 38th character on BabelStone's First Batch Simplified Characters table:


DJ‑38  U+05B66     学   学   學   xué

So it stands to fit in our timeline here.
Funnily enough the same character can also be found in the BabelStone proposal under 3.2

Table of Singapore Simplifications
The table below lists only those 77 characters in the 1969 table of 502 simplified characters that differ from standard PRC simplifications (see Appendix D). The reference number is prefixed “SJ” for “Singapore jiǎntǐzì”). Characters that are proposed for encoding are given a Unicode number of U+XXXXX and printed in red. Unicode points marked with an asterisk are for CJK-D characters, and as of writing are still provisional.



SJ‑26  U+2B52D*      开   開   kāi     Also DJ‑89

This, in all likely-hood, has nothing to do with Nankai - a university established 1919 in Tianjin.

Slightly unrelated: Another funny thing is that these &quot;aborted&quot; characters can sometimes also be seen from the Second Round of Simplified Chinese Characters .
There's this fun blog post from Sinoglot called Park that simplification - but it seems that it hasn't been maintained very well since it was posted in '10 seeing that the original image is now missing, which mentions:

I stumbled across this No Parking sign*…
IMAGE_300 [can't load]

here's probably what it looked like:


… 禁止仃车 (jìnzhǐ tíng chē), in which the third character is 仃 instead of the proper 停, I immediately assumed it was an uneducated mistake propagated by one of society’s fringe characters.
But it turns out to be a fringe character of a different sort.
I asked Sinoglot’s Kellen Parker if he’d run across 仃 used this way. where is my ip ask edd Not only had he run across it, he had the Wikipedia reference! According to that article (which has some fascinating-looking references — I just expanded my Amazon wishlist) 仃 was used for a few years as a completely legitimate simplification of 停 in the so-called “second round” of character simplifications. The simplifications didn’t stick and were repealed in 1986, but it’s possible, if our No Parking writer was educated between 1977 and the mid-80s, that 仃 was simply how they learned to write it.

There is this image in one of the comments though:

with the comment:

I snapped this down the street from where I live. One second-round character (亍 for 街) and one non-standard simplification (柚 for 楼). The community across the street has attractive building signs with the inside of 源 written as 元, which you’d think would be a second-round character, except that the standard was in no way systematic: 原 became [厂+元], but 源 was overloaded on 沅.
I think I first became aware of the characters when I noticed a simplified version of 部 (卩 whose top stroke extends further to the left) cast into some of the iron benches in Jilin’s Longtanshan Park. Or maybe when a friend wrote 量 as 旦 over 力. Fun stuff.

another comment wrote:

The ones I’ve seen most commonly in the wild are 歺 (on hand-lettered breakfast stall signs) and 伩 (in a handwritten note from someone who teaches at Beida). I’d be curious to know how many of these existed previously as folk simplifications: I have no problem believing in 伩 and 歺 (餐 really is a pain in the ass to write by hand when you’re hungry and all you want is some cheap and cheerful 歺); 亍 existed previously as part of 彳亍. 旦 over 力 is just weird.

and another interesting one:

Great post. I am actually doing research on “Second Round Simplifications”. My favorite is 氿. Even as a Mandarin learner rather than a native speaker, this character (meaning of course good old 酒) looks just absurd to me.

A second Sinoglot post Wut if ur kid’s skool thot this wuz fine spelling? attacks this in more detail - but the pics are missing (again!). Shame.
Wikipedia's page of Second Round of Simplified Chinese Characters has quite a bit of explanation too, of how it began:

In 1975, a second round of simplifications, the Second Scheme, was submitted by the Script Reform Committee of China to the State Council for approval. Like the First Scheme, it contained two lists, one (comprising 248 characters) for immediate use and another (comprising 605 characters) for evaluation and discussion.9 Of these, 21 from the first list and 40 from the second also served as components of other characters, which caused the Second Scheme to modify some 4,500 characters.10 On 20 December 1977, major newspapers such as the People's Daily and the Guangming Daily published the second-round simplifications along with editorials and articles strongly endorsing the changes. Both newspapers began to use the characters from the first list on the following day.10

and ended:

The Second Scheme was received extremely poorly, and as early as mid-1978, the Ministry of Education and the Central Propaganda Department were asking publishers of textbooks, newspapers and other works to stop using the second-round simplifications. Second-round simplifications were taught inconsistently in the education system, and people used characters at various stages of official or unofficial simplification. Confusion and disagreement ensued.[11]
The Second Scheme was officially retracted by the State Council on 24 June 1986. The State Council's retraction also emphasized that Chinese character reform should henceforth proceed with caution, and that the forms of Chinese characters should be kept stable.[12] Later that year, a final version of the 1964 list was published with minor changes, and no further changes have been made since.6

Sometimes these things caught in &quot;time warps&quot; are easy to come out.