Learn English – An alternative term for ‘lesser time’

Generally, a term is more rigid and finite than a period.

term:

2a : a limited or definite extent of time; especially : the time for which something lasts : duration, tenure [term of office] [lost money in the short term]

Examples: academic term, term paper, term life insurance, to carry a baby to term

period:

a : a portion of time determined by some recurring phenomenon

Edit to address OP's question edits:

Is it possible to use one or both of them when we describe a point in time (We have time till 1st of December so we have to finish the project within this term/period)

You could say we have a December 1st deadline, so we have to finish the project within this period (between now and December 1st). What you will often see instead of period here is time frame:

: a period of time especially with respect to some action or project

The second part of your question:

and when we describe duration (We have two weeks so we have to finish...)?

...within this period or ...within this time frame are a better fit than term.

The gender issue is a tricky one. Many words ending with -man (e.g., chairman, fireman, layman) seem to be referring to males. As women moved into these professions, some took offense to a word implying that women were not included.

There's been a move away from that, and the -man suffix is often interpreted to mean a person, not necessarily a man. NOAD, Collins, M-W, and Macmillan all define layman with terms like someone or a person, such as:

• a person who does not belong to a particular profession or who is not expert in some field (M-W)
• a person without professional or specialized knowledge in a particular subject (NOAD)

Collins adds a note that layperson is a gender-neutral version of the word (although its definition reads gender-neutral):

a person who does not have specialized or professional knowledge of a subject

CDO doesn't even mention this usage of the word under its British English definition (only listing the religious meaning of the word), but its American English definition does define layman as a non-expert. Like the other dictionaries I've cited, their definition reads gender neutral (a person who is not trained in or does not have a detailed knowledge of a particular subject), although their entry also lists layperson and laywoman as alternate forms.

As for layman vs. user; I've heard the term expert user used to reference to proficient users of a system:

expert user: A person with sufficient knowledge and experience to be able to use a library or computer system effectively and efficiently, with only occasional assistance. The opposite of novice. (ODLIS)

So, perhaps novice is a better word than layman for describing users who, in the context of that system, only have the proficiency of a layman.

So, to recap:

Is layman a gender specific term or can it be used like college 'freshman'?
That can go either way; it can be used gender-neutrally, or you can use layperson instead.

Is it still appropriate to use, or has it been superseded by some other term?
Yes, the word is still in use.

Specific to IT, it seems more appropriate to say user instead of layman. However are the two equivalent or are there any subtle differences?
No, they are not equivalent (for one, not all users are laymen). You might use the term generically, to describe those who aren't "power users", but there are probably better and more clear ways to make that distinction.