Mater respondet se in foro panem invenire sperare.

Because se is a reflexive pronoun that refers back to the subject of the main verb (mater). The mother is speaking about herself, so Latin uses se (accusative) = herself.

• se refers to mater (the subject of respondet)
• eam would mean her referring to some other woman (not the subject)

Because se is the subject of an infinitive (invenire) inside an indirect statement. In Latin, the subject of an infinitive is normally put in the accusative (the accusative + infinitive construction).

It’s an indirect statement (often called accusative and infinitive, or AcI). After many verbs of saying/thinking/perceiving (including respondet), Latin commonly reports what someone says by using:

• an accusative “subject” (se)
• • an infinitive verb (invenire)

So respondet se ... invenire = she replies that she ... finds / will find ...

Latin can use an indirect statement after verbs like respondere when the reply is reported rather than quoted directly.

• Direct style could be something like: Mater respondet: panem in foro inveniam/sperabo. (exact wording varies)
• Indirect style: Mater respondet se ... (reported content)

sperare is another infinitive that depends on respondet and expresses what the mother “replies” about: she replies (that) she hopes...
Then invenire depends on sperare: to hope [to find].
So the structure is:
respondet → se ... sperare → in foro panem invenire

Because Latin is stacking infinitives in a natural way:

• hope
  • to find (English also uses this pattern)
    In Latin, sperare can take an infinitive complement (invenire) just like English to hope to do something.

foro is ablative singular of forum. With in meaning in/within (location, not motion), Latin uses in + ablative.
So in foro = in the marketplace / in the forum (a place where she expects to find bread).

panem is the direct object of invenire (to find). Invenire is a transitive verb, so what is found goes in the accusative.

In an indirect statement, the tense of the infinitive is mainly relative to the main verb:

• present infinitive = action happening at the same time as the main verb (or viewed as a general/ongoing idea)
  Here it fits because she’s expressing a current hope about finding bread. English often uses will in these contexts, but Latin can still use the present infinitive after a present-tense reporting verb (respondet).

Yes, but it would shift the style. Latin often prefers AcI with verbs of saying/thinking. Another option for expressing hope about a future outcome could involve different constructions (especially in more complex sentences), but se ... sperare ... invenire is a very standard, straightforward way to say it.

Latin word order is flexible and often puts key ideas earlier for emphasis or clarity. Here:

• se comes early to mark the start of the indirect statement
• in foro is placed before panem and invenire to set the scene first
• invenire comes before sperare because invenire is the action being hoped for, and sperare wraps up the thought (to hope to find)
  But many other orders could also be grammatical.