can an individual rep force a floor vote on moving his/her proposed act forward?
No. This is why control of the chambers matter. The Speaker of the House and the Majority Leader of the Senate control whether a given bill receives a floor vote. There may be particular circumstances when a member or members can force a vote, but this is not a generic power applicable to all bills.
In the House, the Rules committee determines how and when a bill will get a floor vote. Other committees may also be able to block (not force) a vote, although the Rules committee could override them by changing the rules. The Rules committee has nine members from the majority party and only four from the minority party.
Can s/he join any committee s/he likes? Just one or more than one?
Not any committee. Some committees are more prestigious and harder to get a slot. That said, they do try to accommodate requests if they can (and actively ask for requests). The majority and minority each get so many slots on each committee and the leadership of each party allocate committee assignments.
Representatives can ask for any committee, but they may not get it.
Some committees, like Ways and Means, are considered to be so busy that that will be the Representative's only committee assignment. Other committees allow for multiple memberships. Virtually every member of Congress will be on multiple subcommittees.
Are there budgets put at the representative's disposal?
As part of the House budget, each Representative gets funds for an office. They are assigned offices in the capitol. In their district, they can use their budget to rent space. They can also hire staffers in either place. A Representative can have multiple district offices if the funds will stretch that far. Large districts that need multiple offices tend to be rural districts, so office space will often be cheaper there.
A Representative may choose a new office by seniority. Of course most more senior Representatives already have nice offices, so it may not be worth switching. This matters when a Representative leaves office, as other Representatives may prefer that office to their current office. So the most senior who wants to switch can. Then that Representative's office becomes available and the most senior who wants it can switch. So on and so forth until everyone has an office.
I don't believe that Representatives can displace other Representatives. I believe that a Representative can only lose an office by stopping being a Representative or by voluntarily switching to another.
Can the single representative force the high(est) ranking officers of the executive branch to explain certain matters, in less or in more depth, not in the context of a house committee?
Not officially. However, since they can force them in the context of a committee, officials may choose to explain things to Representatives that they would not choose to explain to others. So not a power per se.
Does the representative have any power relative to state-level officials or bodies?
Not separate from legislative power. In the US, the federal and state governments are separate entities. The state legislatures used to have the power to appoint Senators, but that was repealed via a constitutional amendment.
Can s/he imbue certain spaces or people with immunity from searches and seizures?
This doesn't send like a Congressional power. Perhaps someone else can come up with some odd way of doing that. E.g. appointing an employee to do something that is protected from search/seizure.
Certainly the Representatives are themselves immune from search/seizure while traveling to and from a session. But I don't know that they can imbue others with that. Their local space may be safe from search/seizure--I'm not sure of the precedents there. I.e. it is possible that their personal immunity may extend to places under some circumstances.
etc.
A member of Congress can nominate people to the military academies to become officers.
A member of Congress may take guests on tours of the legislative buildings.
A member of Congress may submit a bill to the Congressional Budget Office and ask them to score it for budgetary impact.
A member of Congress may use the franking power to send mail. I.e. instead of a stamp, they can sign the envelope where a stamp would go.
I do not believe this to be an exhaustive list.
